Preheat oven to 350 F.
Arrange the tomatoes in a single layer on a clean dish towel or layers of paper towels. Sprinkle evenly with the salt, and gently press another dish towel or paper towels on top of the tomatoes. Let sit like this until you’re ready assemble the tart.
Combine the flours, butter, and Parmesan (except for a tablespoon or so - set that aside to use later) in the bowl of a food processor and pulse 20-30 times, until the mixture is sandy and the butter is pea-sized. With the motor running, drizzle in the ice water and blend just until combined. When you press some of the dough between your fingers, it should stick together.
Turn the dough into a 9 or 10-inch tart pan. Working quickly, press the dough into the pan, pressing it into an even layer across the bottom of the pan and up the sides, forming a rim. Chill in the refrigerator for 15 minutes.
Poke the bottom of the crust in several places with the tines of a fork. Cover the tart with a round of parchment, then fill it with pie weights (or dried beans). Bake 15 minutes. Remove from oven and gently remove the parchment and pie weights. Return the tart shell to the oven and continue to bake for another 10 minutes, until deep golden brown. Sprinkle the tart shell with the remaining tablespoon of grated Parmesan. Let sit and allow to cool to room temperature.
Just before serving, arrange the tomatoes in the tart shell. Drizzle with the olive oil and sprinkle evenly with the basil. Slice and serve at room temperature.