Combine onion and mushrooms in a kitchen towel; twist tightly to extract as much moisture as possible. Melt butter in a skillet; add onion and mushrooms. Cook, stirring occasionally, until onion is transparent, about 2 minutes. Add salt and pepper; sauté over high heat, stirring constantly, until liquid has almost cooked away, 4 to 7 minutes. Pour in wine; cook over high heat, stirring constantly, until liquid has cooked away. Stir in parsley.
• 1 tsp. finely minced fresh sage, or more
Heat the oven to 400 degrees. Line a baking sheet with a silicon mat or parchment paper. Mix Parmesan with sage and red pepper. Spoon in 1 tablespoon mounds onto the sheet; press lightly to flatten. Bake until the cheese is melted and browned, 7 to 8 minutes. Remove from oven; cool. Transfer the crisps to a rack.