Place the tomato slices on a double layer of paper towels and sprinkle with salt. Let them stand for at least 30 minutes to drain — this step is vital to prevent the tart from becoming watery.

Pulse the flour, sugar, and salt in a food processor until combined. Add the chilled butter and pulse until the mixture resembles coarse meal, then drizzle in the ice water and pulse until the dough just begins to clump together.

Turn the dough onto a floured surface, form into a disk, wrap in plastic, and refrigerate for at least 1 hour. Chilling the dough ensures it won't shrink during baking.

Roll the dough into a 12-inch circle on a floured surface, then transfer to a 9-inch tart pan with a removable bottom. Press the dough into the corners, trim the overhang, and bake at 375°F with pie weights for 20 minutes. Remove the weights and bake until the crust is golden brown, about 10 more minutes.

Combine the Parmesan, ricotta, mozzarella, and olive oil in a medium bowl, and season with salt and pepper to taste. Spread the cheese mixture evenly over the bottom of the warm, baked tart shell.

Blot the tops of the tomatoes dry with paper towels and arrange them over the ricotta mixture. Bake until the tomatoes are slightly shriveled and the cheese is bubbling, about 20 to 25 minutes.

Let the tart cool on a wire rack for 10 minutes to allow the filling to set. Garnish with basil and serve warm or at room temperature.