Process flour, sugar, and salt together in a food processor until combined. Scatter butter pieces over top and pulse until mixture resembles coarse cornmeal, about 15 pulses. Add 2 tbsp ice water and continue to process until large clumps of dough form and no powdery bits remain, about 5 seconds. If dough doesn't clump, add remaining 1 tbsp water, ½ tbsp at a time and pulse until dough is one big clump, about 4 pulses.

Sprinkle walnut-size clumps of dough evenly into 9-inch tart pan. Working outward from center, press dough into an even layer, sealing any cracks. Working around edge, press dough firmly into corners of pan with fingers. Go around edge once more, pressing dough up the sides of the pan. Use your thumb to level off any overhang at the top. Use excess dough to patch any holes. Lay plastic wrap over dough and smooth out any bumps using the palm of your hand. Leaving plastic on top of dough, place tart pan in the freezer, and freeze until firm, about 30 minutes.

Adjust oven rack to middle position and heat oven to 375 degrees. Set tart pan on a large baking sheet. Press double layer of aluminum foil into frozen tart shell and over edges of pan and fill with pie weights. (If no pie weights, use dried beans, rice or add a third layer of aluminum foil and fill with coins). Bake until tart shell is golden brown and set, about 30 minutes, rotating baking sheet halfway through baking. Transfer baking sheet to wire rack and carefully remove weights and foil. Let tart shell cool on baking sheet.

Spread sliced tomatoes out over several layers of paper towels, sprinkle with ½ tsp salt, and let drain for 30 minutes. Combine 2 tbsp olive oil and garlic in a small bowl. In a separate bowl, mix together Parmesan, ricotta, mozzarella, and remaining 1 tbsp oil, and season with salt and pepper to taste.

Heat oven to 425 degrees. Spread ricotta mixture evenly over bottom of tart shell. Blot tomatoes dry with paper towels and shingle them attractively on top of the cheese mixture in concentric circles, starting at the outside and working inward. Drizzle with garlic-oil mixture. (Tart can be held at room temperature for up to 2 hours before baking.)

Bake tart on a rimmed baking sheet until bubbling and tomatoes are slightly wilted, 20-25 minutes, rotating baking sheet halfway through baking.

Let tart cool on baking sheet for at least 10 minutes or up to 2 hours. Sprinkle with basil and serve.