Place a steamer basket into a pot and fill with water to just below the bottom of the steamer. Bring water to a boil. Add broccoli, cover, and and reduce heat to medium-low. Steam until tender but still crisp, 4 to 5 minutes.
Preheat oven to 425 degrees F (220 degrees C). Coat a 3-quart baking pan with cooking spray.
Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally until tender yet firm to the bite, about 8 minutes. Drain.
Stir ricotta cheese, egg, Parmesan cheese, and parsley together in a bowl.
Puree red peppers in a blender until almost smooth.
Melt butter in a large skillet over medium heat. Add flour and garlic; cook and stir until thickened, about 1 minute. Stir milk and pureed peppers in gradually; cook and stir until sauce is thick and bubbly, 7 to 10 minutes. Add salt, basil, and pepper.
Spread 3/4 cup of the sauce into the bottom of the prepared baking pan. Arrange 3 noodles on top, 1/4 of the ricotta mixture, 1/4 of sausage slices, 1/4 of the broccoli, and 1/4 cup Monterey cheese. Repeat layering 3 more times with the remaining noodles, ricotta mixture, sausage slices, broccoli, and Monterey cheese. Pour the rest of the sauce of top. Cover pan with aluminum foil.
Bake in the preheated oven until heated through and lightly browned, about 20 minutes. Uncover pan and sprinkle the remaining 1/2 cup Monterey cheese and Parmesan cheese over the lasagna. Continue baking until top is bubbling, about 10 minutes more.