Prep ingredients: Bring a large saucepan of salted water to a boil; cover and keep warm over low heat. Grate Parmesan. Trim stem end from eggplant, then cut into 1⁄2-inch pieces. Roughly chop tomatoes. Grate 2 large garlic cloves. In a small bowl, toss panko and 2 tablespoons of the Parmesan to combine. Pick basil leaves from stems and tear large leaves.
Toast breadcrumbs: Heat 2 tablespoons oil in a large skillet over medium-high. Add panko mixture and cook, stirring, until golden, 3-5 minutes. Add half of the garlic. Cook, stirring, until fragrant and breadcrumbs are crisp, about 1 minute. Return breadcrumbs to bowl; season with a pinch of salt. Wipe out skillet.
Make sauce: Add 1⁄4 cup oil to same skillet and heat over medium-high. Add eggplant, 1⁄2 teaspoon salt, and a few grinds pepper. Cook, stirring occasionally, until eggplant is browned and tender, 5-7 minutes. Add tomatoes, remaining garlic, 1⁄2 teaspoon salt, and 1⁄2 cup water. Simmer sauce, breaking up tomatoes with a spoon, until thickened, 5-6 minutes.
Prep dressing & lettuce: While sauce simmers, in a large bowl, whisk vinegar, 2 tablespoons oil, a pinch each sugar and salt, and a few generous grinds pepper. Cut romaine crosswise into 1⁄2-inch pieces, discarding end.
Cook ravioli: Return water to a boil and add ravioli. (If stuck together, gently pull apart only if possible without tearing. Boiling water will help separate.) Cook, stirring gently, until al dente, 3-4 minutes. Drain ravioli, then add to skillet with sauce.
Finish & serve: Place skillet over medium-high heat, add remaining Parmesan in large pinches to avoid clumping. Toss and cook 1 minute. Add half of the basil to sauce and season to taste with salt and pepper. Serve pasta topped with breadcrumbs, remaining basil, and another drizzle of oil. Toss romaine with dressing and serve alongside. Enjoy!