In a large skillet, saute mushrooms and onions in butter until tender. Transfer to a large bowl; cool. Stir in the rice, milk and eggs. Combine flour, baking powder and salt; gradually stir into rice mixture just until combined.
Heat 2 tablespoons oil in the same skillet over medium heat. Drop batter by 2 tablespoons into oil. Fry in batches until golden brown on both sides, using remaining oil as needed. Drain on paper towels.