Rinse mushrooms. Brush cup side of caps with half the vinegar.
Broil, cups up, on rack in a pan 4 to 6" from heat until juicy, 6 to 7 minutes. Turn mushrooms over, brush them with remaining vinegar, and broil until browned, 5 to 6 minutes.
Meanwhile, in a 10 to 12" skillet stir shallots in butter 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.