In a 10- to 12-inch nonstick frying pan, stir mushrooms, onion, and garlic over medium-high heat until mushrooms begin to brown, 8 to 10 minutes.
Meanwhile, rinse chicken and pat dry. Place breasts in a 9- by 13-inch baking dish and bake, uncovered, in a 375° oven for 10 minutes.
When the mushroom mixture is done, remove it from the heat. Stir in the mustard, capers, vinegar, hot sauce, and mayonnaise.
Spread the mushroom mixture over the chicken and continue baking until chicken is no longer pink in center of the thickest part (cut to test), 15 to 20 minutes longer. Transfer to plates and garnish with parsley sprigs. Add salt to taste.