Melt butter in a saucepan over medium heat. Cook and stir mushrooms and shallot in melted butter until the mushrooms are lightly browned and soft, 5 to 7 minutes. Remove saucepan from heat and set aside to cool for 10 minutes.

Transfer cooled mushroom mixture to a blender; add pine nuts, olive oil, vegetable broth, garlic, lemon juice, salt, and pepper. Pulse blender a few times before blending until mixture is finely ground, scraping down the sides of the pitcher with a spatula as needed; scrape into a mixing bowl. Stir Parmesan cheese through the blended mixture.