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Preheat oven to 400 degrees F.
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Line a baking sheet with parchment paper. Lay two sheets of phyllo dough across the parchment, overlapping slightly in the middle to form a large rectangle. Keep the remaining phyllo covered with plastic wrap or wax paper and a damp kitchen towel to prevent in from drying out, which happens quickly.
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Lightly brush or mist the surface of the phyllo sheet with oil. Sprinkle with 1/4 teaspoon bread crumbs. Repeat with the remaining sheets of phyllo, bread crumbs, and olive oil.
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Brush or mist the final sheet of phyllo. Form a crust by folding about 3/4 inch of each side toward the center.
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Spread the pesto evenly over the surface of the tart. Sprinkle with about half the goat cheese. Arrange the tomato slices over the pest and goat cheese and then sprinkled with salt, pepper, and remaining cheese.
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Bake until the crust turns brown and crispy, about 30 minutes. Let cool in the pan on a wire rack about 5 minutes. Scatter basil leaves over the top of the tart and serve warm or at room temperature.