Position an oven rack in the upper third of the oven and a second oven rack in the lower third of the oven. Preheat the oven to 400°F. Line 2 baking sheets with parchment paper. Line another baking sheet with paper towels or a clean kitchen towel.

In a medium bowl, mix together the ricotta, pesto, ¼ cup of the parmesan cheese, and ¼ teaspoon of the salt. Use a sharp knife or a mandoline to slice the zucchini and yellow squash into ⅛-inch-thick slices. Spread the vegetables on the towel-lined baking sheet and sprinkle with the remaining ½ teaspoon salt.

On a work surface, unfold the puff pastry sheets and cut each of them lengthwise into 3 rectangles. Place the pastry on the prepared baking sheets. Leaving a ¼-inch border, prick the center of the puff pastry rectangles with a fork and brush the border with the beaten egg.

Spoon about 3 tablespoons of the ricotta mixture onto each puff pastry sheet, spreading within the border. Pat dry the zucchini and yellow squash and shingle them on the tarts. Nestle the tomatoes around the vegetables, then drizzle the tarts with olive oil. Sprinkle with the pepper and remaining ¼ cup parmesan.

Bake until the puff pastry is golden brown, 18 to 20 minutes, rotating the pans and swapping them between the upper and lower racks after 15 minutes. Allow the tarts to cool for 5 minutes on the baking sheets before moving them to a wire rack.

Sprinkle with the basil before serving.