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Preheat the oven to broil.
Heat a large skillet with high sides to a medium heat. Season chicken breasts with plenty of salt and a little bit of pepper on both sides. Once the skillet is hot, add 1/2 tablespoon of butter and olive oil. Swirl to coat the pan. Sear chicken breasts until golden brown on both sides, about 3-4 minutes per side. Remove the chicken from the skillet and on to a platter or plate. Cover with foil.
Add remaining two tablespoons of butter to the skillet. Once the butter melts, add the onions. Stir to coat the onions. Add salt and sugar. Stir again. Turn the heat down to medium-low and continue to slowly caramelize the onions. Stir every few minutes. After about 15 minutes, the onions should be golden brown and caramelized.
Stir in the cognac and white wine. Bring the mixture to a boil and then reduce to a simmer. Simmer for 2-3 minutes until the alcohol is cooked off and slightly reduced. Stir in flour. Cook for 1 minute.
Slowly whisk in beef broth. Add thyme. Bring to a boil and reduce to a simmer. Simmer for 3-4 minutes until slightly thicked. Add chicken back to the skillet and nestle in the sauce. Cook for another 2-3 mintues until the chicken is almost all the way cooked through. Evenly divide the cheese between the four chicken breasts. Place in the oven on the middle rack. Broil untilt the cheese is melted, brown and crisp.