Sprinkle chicken evenly with 1 teaspoon of the salt and 1/2 teaspoon of the pepper. Heat oil and 1 tablespoon of the butter in a large skillet over high. Add chicken to skillet; cook until golden brown on each side and a thermometer inserted into thickest portion of meat registers 165°F, about 4 minutes per side. Transfer chicken to a plate; set aside. Do not wipe skillet clean.
Reduce heat under skillet to medium-high. Add mushrooms and remaining 1 tablespoon butter; cook, stirring occasionally, until mushrooms release liquid and begin to brown, about 5 minutes. Add shallot; cook, stirring often, until tender, about 3 minutes. Add brandy; cook, undisturbed, until liquid is almost evaporated, about 15 seconds. Add broth, cream, lemon juice, mustard, and Worcestershire; cook, stirring often, until sauce reduces slightly, about 3 minutes. Stir in parsley and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Remove from heat. Nestle chicken and any accumulated juices into mixture in skillet. Serve with lemon wedges.
