1/2 cup finely grated pecorino Romano (about 4 ounces)
Spread the eggplant slices, zucchini strips, and squash strips on paper towels on your work surface; sprinkle with half of the salt. Turn the vegetables and sprinkle with the remaining salt. Set aside for 30 minutes, then rinse the vegetables and pat dry with fresh paper towels.
Position the rack in the center of the oven and heat the oven to 400 degrees Fahrenheit. Lightly oil the inside of a 9-inch square baking dish. Warm the oil in a large skillet set over medium heat. Add the onion and bell pepper; cook, stirring often, until very soft but not brown, about 7 minutes. Add the garlic and cook until fragrant, about 20 seconds.
Stir in the tomatoes, basil, and rosemary. Bring to a simmer, then reduce the heat to low and cook, uncovered, until the consistency of a thick sauce, about 15 minutes, stirring often.
Spread half the tomato sauce in the bottom of the prepared baking dish. Top with even layers of the eggplant rounds, zucchini strips, and summer squash in that order. Spread the remaining tomato sauce on top and sprinkle with the grated cheese.
Bake until bubbly and lightly browned, about 45 minutes. Cool on a wire rack for 10 minutes before serving.