Preheat oven to 450°F. Brush both sides of polenta slices very lightly with oil and arrange on 2 rimmed baking sheets. Roast polenta until lightly golden brown and warmed through, about 25 minutes.
Meanwhile, melt butter with remaining 2 Tbsp. oil in large, deep skillet over medium-high heat. Add mushrooms and cook, stirring often, until tender and browned, about 12 minutes. Add shallots, vinegar, salt, black pepper, and red pepper and cook, stirring, until tender, about 5 minutes. Stir in parsley.
Divide mushroom mixture between baking sheets with polenta. Top with cheese. Continue to roast until cheese is melted, about 5 minutes.
Arrange polenta bites on a platter and serve.
Mushroom mixture can be made 1 day ahead. Transfer to an airtight container and chill.