Special equipment: An 8-inch-square baking dish
For the dough: Combine the flour, Parmesan, sugar and salt in a food processor and pulse until mixed. Add about one third of the chilled butter and process until thoroughly combined (the mixture will be slightly yellow). Add the remaining chilled butter and pulse until the mixture resembles very coarse meal with pea-size bits of butter. Add the vinegar in a single pulse. Gradually add the ice water through the feed tube pulsing just until evenly combined. Squeeze a handful of the dough together, it should just hold its shape and be a little crumbly. If still very powdery, pulse again, adding up to 2 tablespoons ice water.
Turn the dough out on a work surface. Pat about two thirds of the dough into a square about 1/2-inch thick. Repeat with the remaining dough. Wrap each piece tightly in plastic wrap. Refrigerate until firm, at least 1 hour or up to 2 days.
Adjust an oven rack to the bottom position and preheat the oven to 400 degrees F. Butter an 8-inch-square baking dish.
For the filling: Toss the tomatoes with 2 teaspoons salt in a large colander set over a large bowl. Let stand, gently tossing about every 10 minutes, until the tomatoes release some of their juices, about 30 minutes. Drain the juices from the bowl, add the drained tomatoes and toss with the brown sugar and the remaining 1 teaspoon salt.
Meanwhile, heat the oil in a large skillet over medium heat. Add the onion and cook, stirring occasionally, until softened and golden brown, about 5 minutes. Transfer the onion to a medium bowl, stir in the thyme and let cool. Add the pepper Jack, mozzarella, parsley, scallions, mayonnaise, 1/4 cup of the breadcrumbs and the garlic to the onions and stir to combine.
To assemble the pie: Let the dough stand at room temperature for a few minutes until slightly softened to make rolling easier. Place the larger piece of dough between two large pieces of floured parchment; roll into a rough 14-inch square (if the dough gets too soft, pop it in the fridge to firm up and then continue rolling). Roll the remaining dough into a rough 12-inch square.
Transfer the larger piece of dough to the prepared baking dish. Gently press the dough along the bottom and up the sides of the dish. There will be overhang on all sides. Sprinkle the bottom with the remaining 1 tablespoon breadcrumbs; top with the cheese mixture spreading it in an even layer. Then add the tomatoes. Dot with the butter. Put the remaining dough over the filling, lining up the edges with the bottom piece of dough as best you can. Pinch the edges together, trim any excess, then fold and crimp to create a tight seal. Brush the top and edges of the crust with the egg and cut decorative slits in the top of the pie.
Bake until the crust is dark golden brown and the filling bubbles out a little through the slits, 1 hour to 1 hour and 10 minutes. (If the crust browns too quickly, cover the pie loosely with foil.) Let cool at least 2 hours before serving.
Tossing the tomatoes with salt helps release some of their juices so you don't have soup inside your pie.
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