Oil two 9x13-inch baking pans; set aside. Heat the oven to 375 degrees F. Bring a large pot of generously salted water to a boil. Cook pasta shells according to package directions, about 10 minutes, then drain; reserve.
Heat the olive oil in a medium sauté pan, then add the onions and sauté over medium-high heat for about 5 minutes until onions are translucent. Add the spinach, remove from the heat and stir until the spinach wilts. Transfer to a large bowl and let cool.
When cooled, add the chicken, ricotta, chevre, Neufchatel, Parmesan, egg, oregano, basil, salt and pepper. Add half of the Asiago cheese and mix well, mashing the chevre and Neufchatel until incorporated. Transfer the mixture to a gallon-sized plastic bag and cut the corner off to make a piping bag. Pipe the filling into each shell, and place filled shells in the prepared pans.
Cover the shells completely with the sauce, and cover the pans tightly with foil. Bake for 40 minutes, then uncover and sprinkle with remaining Asiago cheese. Bake for 15 minutes longer to melt and brown the cheese.
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