Prep the chicken: Pat the chicken dry and pound to an even thickness (about 1/2–3/4 inch) for even cooking. Season both sides with salt, pepper, and garlic powder.
Sear the chicken: Heat olive oil in a large skillet over medium-high. Add the chicken and cook 4–5 minutes per side until golden and almost cooked through.
Transfer to a plate; tent loosely with foil.
Sauté the veg: Reduce heat to medium. Add butter to the same pan. Stir in the mushrooms and onion with a pinch of salt.
Cook 6–8 minutes, stirring occasionally, until mushrooms release moisture and brown.
Add aromatics: Stir in minced garlic and thyme; cook 30 seconds until fragrant.
Build the sauce: Stir in Dijon and Worcestershire. Pour in the chicken broth and scrape up any browned bits. Simmer 2–3 minutes to reduce slightly.
Thicken lightly (optional): Stir in the cornstarch slurry and simmer 30–60 seconds until the sauce just coats the back of a spoon.
Finish the chicken: Return chicken to the pan, nestling it into the mushrooms.
Top each piece with a slice of Swiss. Cover and cook on low 2–3 minutes until the cheese melts and the chicken reaches 165°F in the thickest part.
Garnish and serve: Taste the sauce and adjust salt and pepper. Sprinkle with parsley.
Serve with lemon wedges if you like a bright finish.