Bring a large pot of water to a boil. Salt water and add pasta. Cook shells 12 to 15 minutes, they should be softened but still undercooked at the center. Drain pasta and cool.
Combine ricotta, 1/2 of the diced mozzarella, a couple of handfuls of Parmigiano and 1/2 of the Asiago. Add parsley to the cheeses and stir to combine.
To a small saucepot over moderate heat add oil, garlic and onions. Saute onions and garlic 5 minutes. Add tomatoes and season sauce with salt and pepper. Simmer sauce 5 minutes and stir in basil leaves
Pour a little sauce into the bottom of a shallow medium sized flameproof casserole dish. Fill shells with rounded spoonfuls of cheese mixture and arrange them seam side down in casserole dish. Top shells with remaining sauce and remaining mozzarella and Asiago cheeses. Place shells in very hot oven or 8 inches from hot broiler and cook 6 to 8 minutes to melt cheeses and bubble sauce.