Melt 2 tablespoons butter in a skillet over medium heat; cook and stir chanterelles in hot butter until tender, 5 to 8 minutes.
Transfer chanterelles to a bowl. Melt remaining 7 tablespoons butter in the same skillet and reduce heat to low. Cook butter until it turns pale brown and gives off a nutty fragrance, about 5 minutes.
Stir tomato and balsamic vinegar into browned butter; cook and stir until tomato is soft, about 2 minutes. Stir cooked tomato and browned butter into mushrooms.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Place scallops onto a broiler pan and broil in oven until just opaque, about 2 minutes per side.
Serve scallops on a platter topped with chanterelles and sauce.
