Coat tops of mushrooms with olive oil. Place mushrooms top-down in a grill-safe dish.
Mix blue cheese, butter, garlic, basil, salt, and pepper thoroughly in a bowl. Scoop evenly into the mushroom caps.
Preheat an outdoor grill for medium-high heat and lightly oil the grate. Cook mushrooms until tender and heated through, about 5 minutes.
Portobello mushrooms would work, too.