1. Preheat oven to 375°.
2. Combine squash, broth, milk, and garlic in a medium saucepan; bring to a boil over medium-high heat. Reduce heat to medium, and simmer until squash is tender when pierced with a fork, about 25 minutes. Remove from heat.
3. Place the hot squash mixture in a blender. Add salt, pepper, and Greek yogurt. Remove the center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Blend until smooth. Place blended squash mixture in a bowl; stir in Gruyère, pecorino Romano, and 2 tablespoons Parmigiano-Reggiano. Stir until combined.
4. Cook pasta according to package directions, omitting salt and fat; drain well. Add pasta to squash mixture, and stir until combined. Spread mixture evenly into a 13 x 9-inch glass or ceramic baking dish coated with cooking spray.
5. Heat oil in a medium skillet over medium heat. Add panko, and cook for 2 minutes or until golden brown. Remove from heat; stir in remaining 2 tablespoons Parmigiano-Reggiano cheese. Sprinkle evenly over the hot pasta mixture. Lightly coat topping with cooking spray.
6. Bake at 375° for 25 minutes or until bubbly. Sprinkle with parsley, and serve immediately.