Pat the chicken breasts dry and season both sides with salt and pepper. Heat olive oil in a large skillet over medium-high heat. Once hot, add the chicken breasts and sear for about 4-5 minutes per side until golden brown and cooked through. Remove chicken from the skillet and set aside.
In the same skillet, reduce heat to medium. Add the thinly sliced onions and a pinch of salt. Cook slowly, stirring occasionally, until the onions turn a deep golden brown and sweet, about 15-20 minutes.
Add the minced garlic to the caramelized onions and sauté for 1-2 minutes until fragrant. Pour in the chicken broth, scraping up any browned bits from the bottom of the skillet.
Stir in the heavy cream, Dijon mustard, and dried thyme. Let the mixture simmer gently for 3-5 minutes until the sauce thickens slightly.
Nestle the cooked chicken breasts back into the skillet, spooning some sauce over the top. Sprinkle the shredded Gruyère cheese evenly over the chicken.
Move the skillet to a preheated broiler for 2-3 minutes or until the cheese is melted, bubbly, and golden brown. Watch carefully to avoid burning.
Remove the skillet from the oven and sprinkle with freshly chopped parsley. Serve immediately with your favorite side.
