Prepare grill for high heat. Lightly score mushroom top. Brush mushrooms and onion with vinaigrette. Place mushrooms and onion on grill. Grill until mushrooms and onion are soft, turning once, about 4 minutes for mushrooms and 6 minutes for onions. Remove mushrooms and onion from grill and place on a plate. Grill buns.
To assemble, spread goat cheese on cut side of each bun. Top bottom bun with one mushroom, onion slice, second mushroom, and spinach. Place top bun on and serve immediately.