Preheat your oven to 375°F (190°C). If using thick chicken breasts, slice them horizontally into thin cutlets or pound them to an even ½-inch thickness so they cook evenly. Season both sides with Italian seasoning, garlic powder, salt, and pepper.
Melt 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Sear the chicken for 2–3 minutes per side until golden brown. Transfer chicken to a plate and set aside.
In the same skillet, add the sliced mushrooms. Cook undisturbed for 3–4 minutes until they begin to brown, then stir in the shallots and garlic. Cook for another 2 minutes until tender. Sprinkle the 1 tbsp of flour over the mushrooms and stir for 1 minute to cook out the raw flour taste.
Pour in the chicken broth, scraping up any browned bits stuck to the bottom of the skillet. Lower the heat to medium-low and whisk in the heavy cream, Dijon mustard, and fresh thyme. Let it simmer for 2 minutes until slightly thickened. Season with a pinch of salt and pepper to taste.
Transfer cream sauce to a 9x13 baking dish, and nest in the chicken. Evenly top each piece of chicken with the shredded Swiss cheese. Transfer to the oven and bake for 12–15 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) and the cheese is melted and bubbly.Turn on the oven broiler for the last 1–2 minutes of cooking to get the Swiss cheese speckled with brown, bubbly spots.
