Bring a large pot of lightly salted water to a boil. Stir in lasagna noodles and return to a boil. Cook the pasta uncovered, stirring occasionally, until the noodles have cooked through but are still firm to the bite, about 8 minutes. Drain and set aside.
Preheat oven to 350 degrees F (175 degrees C).
Spray a 9×13-inch baking dish with cooking spray.
Heat olive oil in a large skillet over medium-high heat. Cook and stir shallots and garlic in the hot oil until shallots are tender and translucent, about 5 minutes.
Stir red bell pepper and red pepper flakes into the shallot mixture until bell pepper is slightly softened, 1 to 2 minutes.
Mix mushrooms into the skillet; season with salt and black pepper. Cook and stir until the mushrooms have given up their liquid and are browned, about 10 minutes more.
Stir ricotta cheese into the mushroom mixture; remove skillet from heat and set aside.
Melt butter in another skillet over medium heat. Whisk in flour until smooth, 2 to 3 minutes.
Whisk milk into flour mixture until smooth; bring to a simmer and whisk constantly until thickened, about 5 minutes. Season with salt and black pepper.
Coat the bottom of the prepared baking dish with a small amount of the white sauce.
Arrange 4 lasagna noodles in a single layer over the white sauce. 1
Spread about 1/3 of the mushroom mixture over the noodles. 1
Pour about 1/3 of the white sauce over the mushroom mixture. Repeat the layers 2 more times, starting with 4 more lasagna noodles. 1
Sprinkle Parmesan cheese, mozzarella cheese, and parsley over the lasagna. 1
Bake in the preheated oven until cheese is melted and sauce is bubbly, 30 to 40 minutes. Allow lasagna to rest for 10 minutes before slicing and serving.