Add noodles to boiling water and cook until pliable and al dente, 7-8 minutes. Drain in a colander and transfer to a plate. Toss gently with 1 Tbsp. olive oil, spread into a single layer, and allow to cool. This makes noodles easier to work with. While noodles boil, cut mushrooms into 1/4" slices. Combine ricotta, basil pesto (reserve 1/2 tsp. for garnish), and half the mozzarella (reserve remaining for topping lasagna) in a mixing bowl.

Line a plate with a paper towel. Heat 1 Tbsp. olive oil in a large pan over medium-high heat. Add mushrooms to hot pan and stir occasionally until mushrooms are browned and slightly softened, 3-4 minutes. Add spinach, 1/2 tsp. salt, and 1/4 tsp. pepper, and cook until just beginning to wilt, 1 minute. Transfer to towel-lined plate to cool. Set aside four mushroom slices to top lasagna. Wipe pan clean and reserve.

Return pan used to cook vegetables to medium heat. Add 1 Tbsp. olive oil and flour. Whisk to combine, then add milk slowly, whisking constantly while adding to avoid lumps. Cook until thick enough to coat the back of a spoon, 2-3 minutes. Don't allow the sauce to boil, or it will separate.

Place casserole dish on prepared baking sheet to collect any drips. Add 1/4 the sauce and a layer of noodles to casserole dish (halve to fit, if needed). Top with half the vegetable mixture, dot surface with half the ricotta mixture, and season with a pinch of salt and pepper. Repeat entire process. Top with a third layer of noodles, then add remaining sauce. Top with remaining mozzarella and reserved mushroom slices.

Spray a piece of foil with cooking spray and cover dish. Bake until bubbly, 25-30 minutes. Remove foil, switch oven to broiler, and broil until top is brown and bubbly, 3-5 minutes. Broiler heat levels vary, so watch lasagna closely to avoid burning!

Top lasagna with reserved basil pesto. Rest 5 minutes before slicing and serving.