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1
To make the garlic mayonnaise: Combine the vinegar and lemon juice into a small bowl. Combine the neutral oil and olive oil in a measuring cup with a spout. Put the 2 egg yolks in a mini food processor. (Alternatively, you can do this by whisking vigorously by hand.) Using a fine Microplane, grate the garlic into the bowl of the food processor with the yolks, add 1 teaspoon kosher salt, and blend for about 30 seconds, until the yolk is pale yellow. (If blade is spinning without catching the yolk, you can add some of the vinegar-lemon juice mixture, bit by bit, until it does.) Add a few drops of the combined oil and pulse to incorporate the oil into the egg mixture. With the machine running and the hole in the lid open, add 2 or 3 tablespoons of the remaining oil a few drops at a time until the egg and oil are emulsified.
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2
Turn off the food processor, take off the lid, scrape down the sides with a rubber spatula, and add a teaspoon of the vinegar-lemon juice mixture. Return the lid and pulse to combine. Continue adding the oil a few drops at a time with the machine running until you've added about half the oil. Stop the machine again and add another teaspoon of the vinegar-lemon juice mixture. Add the remaining oil in a slow steady stream with the machine running constantly, stopping when the mayonnaise thickens to add the remaining vinegar-lemon juice mixture and pulse to combine. If at any point the mixture gets too thick, you can thin with a bit of water. If you run into any trouble with mayonnaise breaking, don't stress! See the video linked in the note above for how to salvage it.
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3
Use the mayonnaise in the egg salad or refrigerate, covered, for up to 3 days. This recipe makes about twice the mayonnaise you'll need for the egg salad-use the other half as a dip or spread (even on egg salad sandwiches).
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4
To make perfect hard-cooked eggs: Bring a large saucepan of water to a boil over high heat. Add 1 tablespoon kosher salt. (The salt is to help the whites solidify quickly if there's a crack in an egg-it doesn't penetrate the shell to season the egg.) Carefully add the 13 whole eggs, lower the heat, and simmer the eggs for 5 minutes. Meanwhile, make an ice bath in a medium bowl and create a bed of paper towels on a plate.
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5
Turn off the heat and let the eggs sit in the hot water for 5 minutes. Remove one tester egg and crack it open. The yolk should be bright yellow and firm but slightly wet in the center. If the yolk is not yet set, leave the rest of the eggs in the hot water for another minute. (Either way, if the tester egg is done to your liking, salt it and eat it as a snack.)
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6
Transfer the eggs to the ice bath to cool completely. Once cool, carefully peel them under running water, and place the peeled eggs on the paper towels to dry.
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7
To make the egg salad: Break the eggs in half to separate the whites from the yolks. Break the yolks in half and drop them into a large bowl. Break the whites into small pieces (about 8 pieces per egg) and drop them into the bowl with the yolks. Sprinkle the eggs with kosher salt, to taste. Add 1/2 cup of the mayonnaise, and aggressively stir the ingredients together using a rubber spatula until combined. (This breaks down the egg yolks, which thickens the mayonnaise and ultimately makes for a creamier egg salad.) Taste and adjust the seasoning. Serve the egg salad or cover and refrigerate for up to 2 days. To serve, transfer the egg salad to a deep serving bowl, piling it in a high mound. Sprinkle generously with the chives.
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8
To make the bagna cauda: Combine the butter, 1/2 cup olive oil, and anchovies in a small saucepan. Using a fine microplane, grate 10 to 12 cloves of the garlic into the pan and cook over medium heat until the anchovies dissolve and the garlic is soft and fragrant, 5 to 6 minutes, stirring constantly so the garlic doesn't brown. Reduce the heat to low and cook the bagna cauda for another 2 to 3 minutes to meld the flavors.
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9
Turn off the heat and let the bagna cauda rest in the pan until you're ready to use it. Serve warm. Stir to recombine the ingredients before serving and from time to time when it's on the buffet or dinner table.
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10
To make the toast: With the oven rack in the middle position, heat the oven to 350u00b0 F. Lay the bread slices on a baking sheet, brush the tops with olive oil, and bake for 15 to 20 minutes, until the bread is golden brown and crispy. Remove the toast from the oven and rub the oiled side of each toast with the remaining garlic clove.
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11
To serve: Give the bagna cauda a stir and spoon 1 tablespoon on each piece of toast. Put the remaining bagna cauda in a tiny saucepan or bowl with a small spoon. Sprinkle a pinch of parsley and a pinch of sea salt on each piece of toast, artfully stack the toast on a small platter or cutting board, and serve alongside the bowl of egg salad.
