Bring a large bowl of 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 slightly firm, about 8 minutes. Drain.
Mix eggs, cottage cheese, ricotta cheese, and Italian seasoning in a bowl; set aside.
Heat olive oil in a skillet over medium-high heat. Cook and stir mushrooms, onion, and garlic in the hot oil until vegetables are tender, 5 to 8 minutes.
Stir flour and black pepper into mushroom mixture until thoroughly combined; add milk. Cook and stir until sauce is thick and bubbly.
Remove skillet from heat and stir spinach and 1/2 cup Parmesan cheese into the mushroom sauce.
Arrange 3 lasagna noodles in a single layer in the prepared baking dish.
Spread 1/3 of the cottage cheese mixture over the noodles.
Spread 1/3 of the mushroom sauce over the cottage cheese mixture.
Sprinkle 1/3 of the mozzarella cheese on top. Repeat the layers twice more. 1
Sprinkle remaining 1/4 cup Parmesan cheese over the lasagna. 1
Bake in the preheated oven until cheese is melted and sauce is bubbly, about 35 minutes. Allow lasagna to rest for 10 minutes before slicing and serving.