Step 1

Heat 1 tablespoon oil in a large ovenproof skillet over medium-high heat. Season chicken with 1/4 teaspoon each salt and pepper and add to the pan. Cook, flipping once, until golden and an instant-read thermometer inserted in the thickest part registers 165°F, about 6 minutes. Transfer to a plate.

Step 2

Reduce heat to medium. Add 2 teaspoons oil and broccoli to the pan. Cook, stirring occasionally, until browned, about 5 minutes. Add water, cover and cook until tender, about 5 minutes more. Transfer to the plate with the chicken.

Step 3

Add the remaining 1 teaspoon oil, onion and mushroom to the pan. Cook, stirring occasionally, until lightly browned, 5 to 8 minutes. Stir in vinegar, Worcestershire, mustard, thyme, the remaining 1/4 teaspoon each salt and pepper and any accumulated chicken juices from the plate. Cook, stirring and scraping up any browned bits, for 1 minute.

Step 4

Meanwhile, position rack in upper third of oven; preheat broiler to high.

Step 5

Return the chicken to the pan and spoon the mushroom mixture on top. Arrange the broccoli around the chicken. Top with cheese. Broil until the cheese is golden brown, 1 to 2 minutes. Serve topped with parsley and/or more thyme, if desired.