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1
Rinse kumquats and lime thoroughly. Cut kumquats in halves to quarters lengthwise, depending on the size. Remove the seeds. Slice the kumquats into eighth inch thickness. Place sliced kumquats in a small, non-reactive bowl. Zest the lime, and add the zest to the kumquats. Juice the lime into the bowl with the kumquats, making certain not to incorporate any seeds. Add salt and 1 cup sugar to the bowl. Stir to combine. Cover and set aside overnight, or up to 24 hours, stirring once or twice during that time.
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2
Place a rack in the lower middle position of the oven, and preheat the oven to 425u00b0F.
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3
Separate one of the eggs, placing the white into a small bowl, and the yolk into a medium bowl. Set the white aside. Add the remaining 3 eggs to the bowl with the yolk. Whisk until foamy. Whisk the melted butter into the eggs in a steady stream. Whisk in the tapioca flour until no clumps remain. Stir the kumquat mixture into the egg mixture. Set aside.
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4
Roll out the bottom crust of your pie dough, and place in a 9-inch pie pan. Trim the top crust so it barely overhangs the edge of the pie pan. Give the filling a final stir, and pour it into the crust. Gently spread out the kumquat pieces.
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5
Add the water to the reserved egg white, and whisk until combined. Brush the egg white mixture onto the outer half-inch or so of the bottom pie crust. (This will help to create a good seal between the top and bottom crusts.)
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6
Roll out the top crust of your pie dough. Gently drape it over the pie. Trim so there is about a half-inch of overhang. Tuck the overhang over, so the edge is between the top crust and the pie pan. Flute the edges to seal. Brush the top of the pie with the egg white mixture, and sprinkle evenly with the remaining tablespoon of sugar. Cut several steam vents in an attractive pattern near the center of the pie.
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7
Bake pie for 30 minutes. Rotate pie 180u00b0 and cover with foil. Decrease oven temperature to 350u00b0F. Continue baking until the top crust has puffed up and a knife inserted into one of the steam vents comes out clean, about 25 more minutes.
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8
Let fully cool on a rack. Can be stored in the refrigerator for several days, but bring to room temperature before serving.
