Cheesy Portobello Chicken Cutlets with Broccoli

Name: Cheesy Portobello Chicken Cutlets with Broccoli
Ingredients: 2 tablespoons extra-virgin olive oil, divided; 4 chicken cutlets (about 1 pound); ½ teaspoon salt, divided; ½ teaspoon ground pepper, divided; 1 large head broccoli, cut into 2- to 3-inch spears; ¼ cup water; 1 cup sliced onion; 1 large portobello mushroom cap, stem and gills removed, sliced; 2 tablespoons balsamic vinegar; 2 tablespoons Worcestershire sauce; 1 teaspoon Dijon mustard; 2 tablespoons chopped fresh thyme, plus more for garnish; 4 ounces sliced Gruyère cheese; Chopped fresh parsley for garnish