Cut one chicken breast in half horizontally through the middle to create two thinner pieces. Repeat with the second chicken breast. Season the chicken with salt and pepper.
Add butter and oil to the skillet and add the chicken. Cook for 3-4 minutes until brown. Turn the chicken over and cook for another 3-4 minutes. Transfer the chicken to a plate and set aside.
Add 2 more tablespoons of butter and the shallots to the skillet. Cook for 2-3 minutes until soft.
Add flour and stir to incorporate with the fat in the pan. Let cook for a minute to cook out the raw flour taste.
Slowly whisk in chicken stock and cream. Add fresh herbs. Cook for 3-4 minutes to thicken. Add the chicken back to the pan along with any juices that accumulated on the plate.
Cover and simmer for 5-7 minutes until the chicken reaches an internal temperature of 165 degrees. The cooking time will depend on the size and thickness of the chicken breasts.
Taste the sauce and season with salt and pepper if needed. Serve with mashed potatoes, rice, or pasta.
If using dried herbs, substitute ⅓ the amount (generally 1 tablespoon fresh = 1 teaspoon dried). For substitutions, try tarragon, parsley, basil, or oregano.
Store leftovers in an airtight container in the refrigerator for up to 3 days. Reheat on low in a covered skillet until warmed through. Individual portions reheat well in the microwave. Slice the chicken before reheating to avoid overheating and drying out the chicken.