Heat 2 Tbsp. of the oil in a large soup pot or Dutch oven over medium heat. Add onion, cook 4 minutes, until softened. Add mushrooms; cook 4 more minutes, stirring occasionally, until golden. Add garlic, tomato paste, oregano, salt, and red pepper flakes; cook 2 minutes, until tomato paste turns brick red and garlic is aromatic. Stir in crushed tomatoes, beans, broth, and 2 cups of water. Bring mixture to a boil over high. Add lasagna noodles, reduce heat, and simmer, uncovered, stirring occasionally, until noodles are cooked, about 12 minutes. Turn off heat and gradually stir in 1/4 cup of the Parmesan cheese.