Prepare the Chicken: Pat each chicken breast dry. Place a piece of plastic wrap on a large cutting board and arrange the chicken breasts on top, spaced evenly. Cover with another piece of plastic wrap.
Pound the Chicken: Use a meat mallet to pound each chicken breast to ¼ inch thickness, then remove the top layer of plastic wrap.
Season the Chicken: Season both sides of the chicken breasts with salt and pepper.
Sear the Chicken: In a large skillet over medium-high heat, add extra-virgin olive oil and 2 tablespoons of butter, swirling to coat the pan. Place the chicken in the skillet and sear for 5-6 minutes per side until golden brown (be careful not to burn; aim for an even golden color). Remove the chicken from the skillet and transfer it to a plate. Cover with tin foil to keep warm.
Cook the Onions: Reduce the heat to medium and add 2 more tablespoons of butter along with the onions. Stir the onions, scraping any brown bits from the bottom of the pan and cook for a few minutes. Add the bay leaf and sugar, stirring occasionally, and cook for 20 minutes until the onions are golden brown. Add the garlic and fresh thyme during the last 3 minutes.
Deglaze the Pan: Pour in the heavy cream and 1 cup of chicken broth, adding a pinch of salt. Deglaze the pan by stirring, and bring the mixture to a simmer.
Thicken the Sauce: In a small bowl, whisk together the flour and the remaining 2 ½ tablespoons of chicken broth until smooth. Add the slurry to the onion mixture, whisking until thickened, and cook for a few minutes.
Serve the Chicken: Remove the bay leaf, return the chicken to the skillet, coat with the sauce, and simmer for another 5 minutes. Serve warm. I love serving this over garlic mashed potatoes—SO GOOD! Enjoy!