Step 1

In a large colander set over a bowl, toss the tomato slices with 1 teaspoon salt; let stand for 20 to 30 minutes, occasionally shaking the colander to encourage liquid to drain. Heat the oven to 425°F with a rack in the middle position.

Step 2

In a food processor, combine the flour, Parmesan, 1 teaspoon of the thyme, ⅛ teaspoon kosher salt and ¼ teaspoon pepper; pulse 3 or 4 times. Scatter about 42 grams (3 tablespoons) butter over the flour mixture and process until the butter is well incorporated, 15 to 20 seconds. Scatter in the remaining butter, then pulse until broken into pieces no larger than small peas, 10 to 12 pulses. Drizzle the ice water over the mixture, then pulse until it forms curdy clumps, 12 to 15 pulses.

Step 3

Turn the dough out onto the counter. Gather and press it firmly into a disk about 5 inches in diameter. Wrap tightly in plastic, smoothing out the edges and forming the disk into a neat round. Refrigerate at least 30 minutes while you roast the tomatoes (or up to 2 days).

Step 4

Shake the colander a final time to drain liquid from the tomatoes. Lay out the slices on paper towels, then pat dry with additional paper towels. Brush a 9-inch glass pie plate with 1 teaspoon of the oil. Shingle the tomatoes in concentric circles, packing them tightly. Sprinkle with ½ teaspoon pepper and 1 teaspoon of the remaining thyme, then drizzle with the remaining 1 tablespoon oil. Bake until softened and most of the liquid has cooked off, 30 to 40 minutes, depending on the ripeness of the tomatoes. Remove from the oven and cool on a wire rack for about 10 minutes; leave the oven on.

Step 5

Remove the chilled dough from the refrigerator. If it is too firm to roll out, let it stand at room temperature for 5 to 10 minutes. Lightly flour the counter, unwrap the dough and set it on the floured surface. Using a rolling pin, roll it into a 10-inch circle, lightly dusting with flour as needed. Lay the dough on the tomatoes, gently pressing it with your hands, then fold and tuck the edges inside the pie plate. Using a paring knife, make slits 1 inch apart all over the dough and all the way through.

Step 6

Bake until the crust is deeply browned and juices are bubbling at the edges, 30 to 35 minutes. Set the pie plate on a wire rack, then run a knife around the edge of the pie plate to loosen the tart. Cool for 30 minutes.

Step 7

Invert a platter onto the pie plate and, holding them together, carefully re-invert. Lift off the pie plate. Serve warm or cool to room temperature. Just before serving, sprinkle with addtional pepper, the remaining 1 teaspoon thyme and kosher or flaky sea salt (if using); lightly drizzle with additional oil.