Heat olive oil and butter in a large nonstick skillet over medium heat. Mix 2 1/2 tablespoons balsamic vinegar, Dijon mustard, and garlic in a bowl and stir chicken breast halves into the mixture to coat. Transfer chicken with marinade into the skillet and cook, turning occasionally, until chicken is no longer pink inside, 5 to 8 minutes per side. Transfer chicken to a platter and keep warm.
Pour 1/3 cup chicken stock into skillet and stir to dissolve browned bits of food from the skillet. Cook and stir shallot and mushrooms until mushrooms are tender, about 5 minutes. Stir in 1/3 cup chicken stock, herbes de Provence, and 1 1/2 teaspoons balsamic vinegar. Cook, stirring occasionally, until mushrooms are deep brown in color, about 2 minutes.
Return chicken breasts to pan and top each with 1/2 slice of provolone cheese. Cover skillet and let cheese melt; serve with mushrooms.