Season both sides of the chicken breasts with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat until shimmering.
Add the chicken breasts to the skillet and cook for 5-6 minutes on each side until golden brown and cooked through. Remove chicken from skillet and set aside on a plate.
Reduce heat to medium. Add butter to the same skillet. Once melted, add the sliced onions and stir to coat in the butter and oil.
Cook the onions, stirring occasionally, for about 15 minutes until deeply caramelized and golden brown.
Add minced garlic, dried thyme, dried rosemary, and smoked paprika to the onions. Cook for 1 minute until fragrant.
Sprinkle the flour over the onions and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Slowly pour in the chicken broth while stirring to prevent lumps. Bring the mixture to a simmer and cook for 3-4 minutes until slightly thickened.
Stir in the heavy cream, Worcestershire sauce, shredded Gruyère cheese, and grated Parmesan cheese. Stir until the cheese is melted and the sauce is creamy.
Return the cooked chicken breasts to the skillet, nestling them into the sauce and onions.
Simmer for an additional 5 minutes over low heat to allow the chicken to absorb the flavors and heat through.
Garnish with freshly chopped parsley before serving.