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You will need 2 baking sheets, a large pastry bag, a 3/4-inch plain tip for piping shells, a 1/4-inch plain tip for filling the eclairs. My filling method makes it completely unnecessary to have the special long eclair tip shown in one photo.
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In a non-convection oven, plan to bake one sheet at a time. Preheat the oven to 400u00b0 F with a rack in the lower third. (For a convection oven, you'll need four baking sheets; adjust the temperature according to your oven's instructions.)
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Make two parchment paper templates the size of your baking sheets as follows: Draw 2 sets of heavy parallel guidelines 4 1/2 inches apart (to make 4 1/2 inch eclairs shells) on each parchment.
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Whisk the whole eggs and the egg white together briefly. Divide the egg mixture into unequal portions-3/4 cup and 1/4 cup-and set both bowls near the stove. (Set aside any extra egg in a different bowl.)
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Put the butter, sugar and salt with water and milk in heavy bottom saucepan 7 to 8 inches in diameter. Bring the liquid to a simmer over medium heat, stirring frequently so that butter is completely melted by the time the liquid simmers. If it seems like the butter will not be melted in time, slide the pot off the burner and stir until the butter is melted, then return the pot to the burner and continue.
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When the liquid is simmering, dump all of the flour into the pot and commence stirring briskly with a wooden spoon or spatula until all of the flour and liquid are blended and form a stiff paste resembling mashed potatoes, with no patches of dry flour. Continue stirring-spreading, and mashing the paste over the bottom of the pan and gathering it back together again-for 4 to 5 minutes to cook the flour and dry the paste sufficiently to absorb the eggs that will be incorporated in the next step. Some crust will form on the sides and bottom of the pot (don't try to scrape and blend it back into the paste)-and your arm will be tired before you are finished.
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Turn the hot paste into the bowl of an electric mixer fitted with a paddle (or a bowl in which you can operate a handheld electric mixer). Mix at low speed for 3 to 5 minutes until the paste has cooled to 103 degrees.
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Beating on medium speed, pour in the 3/4 cup of eggs in a slow, steady stream. Scrape the sides and bottom of the bowl as necessary, and if the mixture does not form a cohesive mass, turn the mixing speed up briefly to bring it together. Once the eggs are beaten in, turn the mixer off. If the batter is glossy and soft enough that it slumps slowly downwards from the beater-or falls very slowly from a big spoon-no more egg is needed. Beat the paste for another 2 minutes at medium speed.
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If the batter is dull and does not slump or fall slowly, judiciously add in as much of the last 1/4 cup of eggs as it takes to meet the criteria described and then beat for the last 2 minutes. (Combine all leftover egg and set aside for egg wash-whisked with a little water until paint-able-and/or breakfast.)
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Form and bake the choux: Set one template upside down on a baking sheet, and the other upside down on the counter. Fit the pastry bag with the large tip and fill with the paste. Pipe 6 or 7 fat fingers, 4 1/2 inches long, on each template, preparing your first and second batch. Use a natural bristle pastry brush to brush the fingers lengthwise with water or egg wash. (Dip the brush and remove excess liquid against the side of the bowl before brushing.) Rake the brushed fingers gently with the tines of a form. (Brushing and raking all help the shells to expand more evenly in the oven-if your piping is not smooth or even, use the brush to even them up.)
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Set the baking sheet with choux on a second baking sheet (turned upside-down if rimmed) and set the ensemble in the oven. (Double sheeting protects the bottoms of the shells from becoming too dark before the shells are fully baked). Bake 15 minutes without opening the oven. Shells should be puffed and golden. Rotate the baking sheet front to back and turn the oven down to 375u00b0 F. Bake 15-18 minutes until the shells are more brown than golden brown (pale shells will be under-baked inside, soft on the outside, and may collapse as they cool).
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Transfer individual shells from the pan to a cooling rack. Turn the oven temperature back up to 400u00b0, and set pans aside to cool.
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While shells are still hot drill two holes in the bottom of each one as follows: insert the tip of a paring knife one inch from one end of a shell and rotate it like a drill until the hole is slightly larger than 1/4 inch. Repeat one inch from the opposite end of the shell. (Holes release steam and are handy for filling later). Let shells cool completely.
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Once the pans are cooled, slide the second drawn-on parchment sheet with the piped choux on it onto one baking sheet, double sheet it by putting it atop the second baking sheet, and repeat instructions above for your second batch of eclair shells.
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When shells are completely cool, you can fill them with cream (https://food52.com/recipes/35141-vanilla-rice-flour-pastry-cream), or, better yet, freeze them for another day. Fresh or frozen, shells can be re-crisped (without thawing) on a baking sheet in a preheated 375 F oven until very hot to the touch, 6-8 minutes. Twice baking is not only an excellent hack whenever shells do not turn out quite as crispy as you want them to be, but might even make the best shells of all!
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To fill eclairs, scrape the pastry cream into a clean, dry, pastry bag fitted with the plain 1/4-inch tip. Cradle a shell in your palm and insert the tip of the bag into one hole. Squeeze cream into the shell until it wants to ooze back out. Repeat with the second hole. Scrape off any cream that has oozed and set the eclair right side up on a tray. Repeat with the remaining eclairs. Chill before glazing.
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Make the glaze: Melt the chocolate with the butter, corn syrup, and water in a stainless steel bowl set directly in a wide skillet of almost simmering water. Stir frequently until the chocolate is almost entirely melted. Remove the bowl from water and stir to finish melting the chocolate. Use immediately or set aside until later.
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To glaze eclairs, cool freshly made glaze (or gently reheat glaze that has set) to 88-90 F and pour it into a container long enough to fit an eclair, such as a small loaf pan, or a wide bowl. Dip the top of each chilled pastry into the glaze. Refrigerate immediately to set the glaze. Store eclairs in the refrigerator. They are best within 8 hours or filling, but still darned good for another day or two.
