Melt 2 tablespoons butter in a skillet; cook and stir green onion until softened, about 2 minutes. Transfer green onion to a bowl. Stir in crabmeat, bread crumbs, 1/2 cup Monterey Jack cheese, egg, lemon juice, and dill weed until well mixed.
Pour 1/2 cup melted butter in a 9×13-inch baking dish; turn mushroom caps in butter to coat. Fill mushroom caps with the crab mixture and sprinkle with remaining 1/2 cup Monterey Jack cheese. Pour white wine into baking dish.
Bake in preheated oven until cheese is melted and lightly brown, 15 to 20 minutes.