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Special equipment: an 8-inch-square baking dish
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For the dough: Combine the flour, Parmesan, sugar and salt in a food processor and pulse until mixed.
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Add about one third of the chilled butter and process until thoroughly combined (the mixture will be slightly yellow).
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Add the remaining chilled butter and pulse until the mixture resembles very coarse meal with pea-size bits of butter.
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Add the vinegar in a single pulse.
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Gradually add the ice water through the feed tube pulsing just until evenly combined.
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Squeeze a handful of the dough together, it should just hold its shape and be a little crumbly.
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If still very powdery, pulse again, adding up to 2 tablespoons ice water.
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Turn the dough out on a work surface.
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Pat about two thirds of the dough into a square about 1/2-inch thick.
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Repeat with the remaining dough.
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Wrap each piece tightly in plastic wrap.
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Refrigerate until firm, at least 1 hour or up to 2 days.
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Adjust an oven rack to the bottom position and preheat the oven to 400 degrees F. Butter an 8-inch-square baking dish.
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For the filling: Toss the tomatoes with 2 teaspoons salt in a large colander set over a large bowl.
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Let stand, gently tossing about every 10 minutes, until the tomatoes release some of their juices, about 30 minutes.
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Drain the juices from the bowl, add the drained tomatoes and toss with the brown sugar and the remaining 1 teaspoon salt.
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Meanwhile, heat the oil in a large skillet over medium heat.
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Add the onion and cook, stirring occasionally, until softened and golden brown, about 5 minutes.
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Transfer the onion to a medium bowl, stir in the thyme and let cool.
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Add the pepper Jack, mozzarella, parsley, scallions, mayonnaise, 1/4 cup of the breadcrumbs and the garlic to the onions and stir to combine.
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To assemble the pie: Let the dough stand at room temperature for a few minutes until slightly softened to make rolling easier.
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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).
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Roll the remaining dough into a rough 12-inch square.
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Transfer the larger piece of dough to the prepared baking dish.
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Gently press the dough along the bottom and up the sides of the dish.
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There will be overhang on all sides.
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Sprinkle the bottom with the remaining 1 tablespoon breadcrumbs; top with the cheese mixture spreading it in an even layer.
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Then add the tomatoes.
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Dot with the butter.
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Put the remaining dough over the filling, lining up the edges with the bottom piece of dough as best you can.
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Pinch the edges together, trim any excess, then fold and crimp to create a tight seal.
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Brush the top and edges of the crust with the egg and cut decorative slits in the top of the pie.
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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.
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(If the crust browns too quickly, cover the pie loosely with foil.)
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Let cool at least 2 hours before serving.