Preparation

Make the crust

• Combine the flour, Asiago, basil, 3/4 tsp. salt, and 1/4 tsp. pepper in a food processor. Pulse a few times just to combine. Scatter in the butter and pulse until peasize clumps form. With the food processor running, slowly add 2 Tbs. of the ice water through the feed tube. Start dribbling in the third tablespoon bit by bit, stopping once the dough forms a ball.
• Turn the dough onto a floured surface and press into a 6-inch disk. Wrap in plastic and chill for at least 30 minutes and up to 1 day.

Make the filling

• In a 4-quart saucepan, warm the oil over low heat. Add the garlic, anchovy paste (if using), and red pepper flakes and cook, stirring, until fragrant and sizzling, 3 to 5 minutes. Add the tomatoes, olives, and capers. Cover and continue cooking over low heat, stirring occasionally, until most of the tomatoes have burst and the liquid has mostly evaporated, about 15 minutes. Uncover and continue to cook, stirring constantly and breaking any tomatoes that have not burst, about 1 minute. Spread on a large dinner plate or tray and cool to room temperature, stirring occasionally.
• Meanwhile, whisk the eggs in a medium bowl. Add the yogurt and 1/4 tsp. pepper, and whisk until smooth. Set aside.

Assemble and bake the tart

• Position a rack in the center of the oven and heat the oven to 425°F.
• On a lightly floured surface, roll the dough into a 12-inch round. Press into a 9-inch fluted tart pan with a removable bottom, folding in any overhang to create double-thick sides. Set the tart pan on a foil-lined baking sheet.
• Sprinkle 1/4 cup of the Asiago over the bottom of the tart. Spread the tomato mixture on top of the cheese. Slowly pour in the egg mixture.
  
  
• Bake for 20 minutes. Sprinkle the remaining 2 Tbs. Asiago on top of the tart and return to the oven until golden, puffed, and set, 15 to 20 minutes longer. Transfer the tart to a wire rack (without the baking sheet), and cool 10 minutes before removing the outer ring of the tart pan. If transferring from the metal base onto a serving plate, cool 5 to 10 minutes longer before sliding a long spatula underneath the tart to ease it off the base. Garnish with the basil. Serve warm or at room temperature.