Heat oven to 400°F. Wipe mushrooms clean with a damp paper towel or pastry brush; trim ends. Remove stems from mushrooms and finely chop stems; set aside. Oil a baking sheet; set aside.
Melt 2 tablespoons of the butter in a medium frying pan over medium heat until foaming. Add mushroom caps and season with salt and freshly ground black pepper. Cook until tender and golden brown, about 5 minutes. Remove caps from the pan and place top down on prepared baking sheet; set aside.
Return the frying pan to medium heat and melt remaining tablespoon of the butter until foaming. Add mushroom stems, season with salt and freshly ground black pepper, and cook until tender and slightly browned, about 5 minutes. Place stems and leftover stuffing in a medium bowl and mix to combine.
Fill mushroom caps with about 2 heaping teaspoons stuffing each. Bake, stuffing side up, until lightly browned and toasted, about 10 minutes.