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.
Melt 2 tablespoons butter in a large pot over medium heat. Cook and stir portobello mushrooms until soft, 5 to 10 minutes. Season lightly with salt and pepper.
Melt remaining 2 tablespoons butter in a large pot over medium heat. Cook and stir onion and garlic in butter until deeply browned, about 25 minutes. Pour in milk and heavy cream; bring to a simmer. Whisk in Parmesan cheese until sauce thickens slightly. Remove from heat; season lightly with salt and pepper.
Preheat oven to 350 degrees F (175 degrees C).
Layer lasagna noodles, portobello mushrooms, onion sauce, feta cheese, and mozzarella cheese in a baking dish, ending with cheese on top.
Bake in the preheated oven until lasagna is heated through and lightly browned on top, about 30 minutes.