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Preheat oven to 400° F.
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Roll puff pastry to desired dimensions on floured surface to fit into either a rectangular tart pan with removable bottom or round tart pan. Line tart pan with pastry; trim pastry. Line pastry with parchment paper filled with dried beans or other pie weights. Bake for 25 minutes in preheated 400° F oven. Remove from oven; remove parchment paper and dried beans. Let cool 10 minutes.
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Meanwhile, slice tomatoes 1/4-inch thick; spread the tomatoes out over several layers of paper towels. Sprinkle with salt and let drain for 30 minutes. Gently blot the tops of the tomatoes dry with paper towels before using. In a small bowl, whisk together 2 tablespoons of the olive oil and garlic; set aside.
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In a separate bowl, mix the ricotta cheese, Mozzarella cheese, Parmesan cheese, and remaining 1 teaspoon olive oil together.
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Reduce oven heat to 375° F.
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Spread the ricotta mixture evenly over the bottom of the cooled tart shell. Shingle the tomatoes attractively on top of the ricotta mixture. Drizzle the garlic and olive oil mixture evenly over the tomatoes.
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Bake the tart until the cheese is bubbling and the tomatoes are slightly wilted, 20 to 25 minutes.
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Allow he tart cool slightly on a wire rack for 5 minutes; sprinkle with the basil. Remove the tart from the tart pan and transfer to a serving platter or cutting board.
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Yield: 4 servings.