Position racks in upper and lower thirds of oven; preheat to 450°F.
Place squash cut-side down in a microwave-safe dish; add 2 tablespoons water. Microwave, uncovered, on High until the flesh is tender, about 10 minutes. (Alternatively, place squash halves cut-side down on a rimmed baking sheet. Bake in a 400°F oven until the squash is tender, 40 to 50 minutes.)
Meanwhile, heat oil in a large skillet over medium heat. Add broccolini, garlic and red pepper (if using); cook, stirring frequently, for 2 minutes. Add water and cook, stirring, until the broccolini is tender, 3 to 5 minutes more. Transfer to a large bowl.
Use a fork to scrape the squash from the shells into the bowl. Place the shells in a broiler-safe baking pan or on a baking sheet. Stir ¾ cup mozzarella, 2 tablespoons Parmesan, Italian seasoning, salt and pepper into the squash mixture. Divide it between the shells; top with the remaining ¼ cup mozzarella and 2 tablespoons Parmesan.
Bake on the lower rack for 10 minutes. Move to the upper rack, turn the broiler to high and broil, watching carefully, until the cheese starts to brown, about 2 minutes.
Easy cleanup: To save time and keep your baking sheet looking fresh, line it with a layer of foil before you bake.