Heat two large skillets over medium-high heat, melt 3 tablespoons of butter in each of the skillets and divide the mushroom caps between the two. Cook and stir the mushroom caps until the edges are slightly soft, about 5 minutes. Place the mushrooms in a colander to drain and cool.
Stir together the cream cheese and goat cheese until well blended. Mix in the onions and mushroom stems. Use all of the filling to generously fill each mushroom cap and place, filling side up, in a baking pan.
Preheat the oven broiler for high heat.
Melt the remaining 1/4 cup of butter with the garlic in a small saucepan over medium heat, cook the garlic for 1 minute once the butter has completely melted. Drizzle the garlic butter over the filled mushroom caps.
Place the pan of mushrooms in the preheated oven to broil until golden brown, about 5 minutes.
