Heat broiler. In a food processor, puree the tomatoes, garlic, 1 tablespoon of the oil, and ¼ teaspoon each salt and pepper.
In 2 batches, arrange the eggplant slices on a broilerproof baking sheet, brush with 2 tablespoons of the oil, and season with ½ teaspoon salt and ¼ teaspoon pepper. Broil until charred and tender, 3 to 4 minutes per side.
Meanwhile, in a small bowl, combine the ricotta, egg, basil, and ¼ teaspoon each salt and pepper.
Spread half the tomato sauce in the bottom of an 8-inch square baking dish. On top of it, layer a third of the eggplant slices and half the ricotta mixture. Repeat with another layer of eggplant and ricotta. Top with the remaining eggplant and tomato sauce. Sprinkle with the Asiago.
Reduce oven to 400º F. Bake the lasagna until bubbling, 15 to 20 minutes. Let rest for 10 minutes before serving.
Divide the greens among plates, drizzle with the remaining tablespoon of oil; sprinkle with ¼ teaspoon each salt and pepper. Serve with the lasagna.