Cook pasta in salted water according to package directions. Drain and cool slightly, about 10 minutes.
While pasta cooks, preheat broiler with oven rack 6 inches from heat. Toss together squash, zucchini, tomatoes, garlic, 2 tablespoons of the oil, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper on an aluminum foil-lined rimmed baking sheet. Spread in a single layer; broil until charred and tender, about 10 minutes.
Preheat oven to 350°F. Stir together ricotta, Romano, basil, vinegar, and remaining 1 tablespoon oil, 1 teaspoon salt, and 1/2 teaspoon pepper in a small bowl.
Combine pasta, cooked vegetables, and marinara sauce in a large bowl; gently stir to combine. Spoon half of pasta mixture into a 13- x 9-inch baking dish. Dollop 1 cup of ricotta mixture evenly on pasta mixture. Repeat with remaining pasta mixture and ricotta mixture. Top evenly with mozzarella.
Bake at 350°F until lightly browned and bubbly, about 25 minutes. Let stand 5 minutes before serving. Sprinkle with fresh basil leaves.