Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, reserving about 3/4 cup of the pasta cooking water.
While the pasta is working, in a large skillet, heat about 11/2 tablespoons EVOO, 1 1/2 turns of the pan, over medium-high heat. Stir in 2 cloves garlic for 1 minute, then add the kale to wilt to crisp-tender, a couple of minutes. Season with salt, pepper and a little nutmeg. Transfer to a plate and reserve the skillet.
Preheat the broiler and position a rack in the upper third of the oven. Place the ricotta cheese in the bottom of a casserole.
Place the reserved skillet over medium heat and add the remaining 2 tablespoons EVOO, 2 turns of the pan. Stir in the remaining 2 cloves garlic for 1 minute, then add the onion and stir for 2 minutes. Using a blender or food processor, puree the tomatoes. Add them to the garlic and onion and season with salt and pepper. Stir in the basil to wilt. Lower the heat to a simmer.
Add the pasta, reserved cooking water and the kale to the ricotta in the casserole; season with salt and pepper. Top with two-thirds of the tomato sauce, a thin layer of parmigiano-reggiano and the mozzarella. Broil for a few minutes to brown the cheeses. Pass the remaining sauce and parmigiano-reggiano at the table for topping.