Combine squash, 1 cup broth, milk, and garlic in a medium saucepan; bring to a boil. Reduce heat to medium; simmer until squash is tender (about 20 minutes). Remove from heat.
Place squash mixture in a blender. Add salt, pepper, and nutmeg. Remove 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; add Gruyère cheese and 5 ounces mozzarella cheese, stirring until cheese melts and mixture is smooth.
Heat a large skillet over medium heat. Add oil to pan; swirl to coat. Add onion and mushrooms; cook 7 minutes or until browned and liquid evaporates. Add chard and remaining 1 tablespoon broth. Cover and cook 2 minutes or until chard wilts. Place chard mixture in a fine sieve; drain 5 minutes. Place chard mixture in a bowl. Add pine nuts; toss to combine.
Spread 1/2 cup squash sauce in bottom of a broiler-safe 11 x 7-inch glass or ceramic baking dish coated with cooking spray. Arrange 3 noodles over sauce; top with half of chard mixture. Dollop ricotta cheese on top of chard. Spread half of remaining sauce over top. Arrange 3 noodles over sauce. Top with remaining chard mixture; top with remaining sauce. Sprinkle evenly with remaining 5 ounces mozzarella cheese and Parmigiano-Reggiano cheese. Cover with foil coated with cooking spray. Bake at 375° for 35 minutes. Uncover and bake an additional 10 minutes or until bubbly.
Preheat broiler to high. (Keep lasagna in oven.)
Broil lasagna 3 minutes or until cheese is golden brown. Remove from oven; let stand 10 minutes.