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For the homemade flatbread dough: Start by making the dough, as it needs 4 hours to rise in a warm environment. Sift the flour into a bowl. Put a little more than 2 tablespoons warm water in a separate bowl and add the kosher salt and yeast. Let sit for 1 minute, then stir until dissolved. Pour the yeast mixture into the flour.
Begin to knead the dough by mixer or by hand until the dough begins to feel elastic, about 7 minutes if using a tabletop mixer or 15 minutes if by hand. (You will know it's ready by poking a hole in it-- if the hole fills in quickly, it's ready.) Put the dough ball into a metal bowl, pat with the olive oil, cover entirely with a large plastic bag (leaving room to rise) and leave in a warm place. Allow to rise until the dough grows 2 1/2 times its original size, about 4 hours (time varies depending on how warm the room is).
For the chimichurri marinade and chicken: Meanwhile, remove the parsley leaves from the stems and add the leaves to a blender (save the stems for making stock or discard). Add the olive oil, white vinegar, black pepper and garlic and blend. Put the chicken in a container and add 3/4 cup of the marinade, coating the chicken entirely (refrigerate the unused marinade for another use). Refrigerate the chicken until ready to use.
For the basil pesto: Remove the basil leaves from the stems and add the leaves to a blender (save the stems for making stock or discard). Add the olive oil, garlic and lemon juice and blend. Set aside.
For the mayonnaise: Place a damp towel underneath a metal bowl. Add the egg yolk to the bowl and whip. Begin to pour in the canola oil very slowly while whisking the egg. When the mixture starts to become too thick, add a touch of vinegar while continuing to whisk. Be careful not to add too much at once and to maintain a steady pace of whisking so the yolk doesn't break. Keep adding the canola oil, vinegar and olive oil until incorporated.
Blend one-third of the basil pesto together with two-thirds of the mayonnaise to make basil pesto aioli (refrigerate the remaining mayonnaise and pesto for another use). If desired, transfer the aioli to a squeeze bottle for easy use. Set aside.
Roll the risen dough out into a sheet 1/4 inch thick. Cut into 6 rectangular flatbreads, each about 8 inches by 4 inches.
Preheat a grill or grill pan.
Grill the chicken until fully cooked but still tender. Cut the peaches into 1/2-inch slices and grill until slightly soft. Grill the flatbread, approximately 30 seconds per side.
To serve: Place a slice of Brie cheese lengthwise along the middle of each flatbread. Add the grilled chicken, covering the cheese so it melts. Apply a generous amount of the basil pesto aioli. Cover with a layer of the spinach, and top with the grilled peaches. Sprinkle with Parmesan if desired.
You might prefer to make a larger quantity of the flatbread and freeze extra for later. To store the flatbread, use plastic wrap and place 2 per sheet, leaving a space between the dough pieces. Transfer the wrapped dough to a plastic container to freeze for long-term storage.