Preheat the grill to 375. Cover the grate with heavy-duty aluminum foil, which you’ve sprayed with non-stick spray.
Melt the butter in a saucepan over the grill or side burner, add the shallots and cook until soft, 3 to 4 minutes. Add bread crumbs, shallots, Stilton, sage, thyme, and garlic, and process briefly until well combined.
With a teaspoon press the stuffing lightly into the mushroom caps. Transfer the mushrooms to the prepared grill and cook on the foil until the mushrooms wilt and the stuffing begins to brown, 14 to 18 minutes.
Remove the mushrooms from the foil with a spatula, sprinkle with paprika or parsley, and serve as an appetizer with a chilled adult beverage.