In a large skillet, heat the olive oil and butter over medium heat. Add the sliced onions and sauté for about 10 minutes, stirring occasionally, until they start to soften.
Sprinkle the brown sugar over the onions. Continue to cook, stirring frequently, for another 15-20 minutes, until the onions are deeply golden brown and caramelized. Remove the onions from the skillet and set them aside.
Season the chicken breasts with garlic powder, salt, and pepper on both sides. In the same skillet, add the chicken breasts and cook over medium-high heat for 6-7 minutes per side, or until the chicken is cooked through and juices run clear.
Once the chicken is cooked, remove it from the skillet and place it in a baking dish. Top each chicken breast with a generous portion of the caramelized onions and slices of Swiss cheese.
Preheat your oven to 375°F (190°C). Bake the chicken for about 10-12 minutes, or until the cheese is melted and bubbling.
Remove from the oven and let cool slightly. Garnish with fresh thyme sprigs if desired before serving.
Notes
Arrange the chicken on a large platter, drizzle with any remaining caramelized onions, and sprinkle with fresh thyme for a touch of color. Serve with a side of roasted vegetables or a light salad.