Preheat the oven to 375 degrees F (190 degrees C). Grease an 8x8-inch baking dish. Add gnocchi, chicken, broccoli, and peppers to the dish.
Combine Alfredo, chicken broth, pesto, garlic, Italian seasoning, salt, pepper, red pepper flakes, and 2 tablespoons Parmesan in a bowl; pour over baking dish and stir to coat all ingredients. Sprinkle mozzarella and remaining Parmesan evenly over the top.
Stir breadcrumbs and olive oil together and sprinkle over cheese. Cover casserole dish tightly with foil.
Bake covered in the preheated oven for 30 minutes. Uncover and bake until bubbly and lightly golden, 10 to 15 more minutes. Rest casserole for 5 to 10 minutes, then add a squeeze of lemon juice and fresh parsley.