Rinse mushrooms. Brush cup side of caps with half the vinegar. Broil, cups up, on rack in a pan 4 to 6 inches from heat until juicy, 6 to 7 minutes.
Turn mushrooms over, brush them with the remaining vinegar, and broil until browned, 5 to 6 minutes.
Meanwhile, stir shallots in butter in a 10- to 12-inch frying pan over medium-high heat until limp, 3 to 5 minutes. Add spinach and stir over high heat until leaves wilt, 3 to 4 minutes.
Place mushrooms, cups up, on plates and fill with spinach.