Season chicken breasts with salt and pepper. Dredge in flour, shaking off the excess. Set aside.
Heat the butter and olive oil in a large pan or skillet over medium-high heat.
When hot, fry chicken breasts until golden on each side and just cooked through, about 3-4 minutes each side, depending on the thickness of your chicken. Transfer to a warm plate.
Heat the remaining 2 tablespoons of butter in the pan. Sauté shallots until soft, about 4 minutes, then add the garlic and cook until fragrant, about 1 minute.
Add the wine to deglaze the pan, scraping up the browned bits from bottom of pan with the end of your wooden spoon. Simmer until reduced to half, about 4 minutes.
Stir the Dijon mustard into the pan juices. Add the chicken broth and herbs. Bring to a boil, then reduce heat to medium. Stir in cream and let simmer until sauce starts to thicken, about 3 minutes.
Add chicken back into the pan. Simmer gently until chicken is heated through and sauce has thickened, about 5 minutes. Taste test and adjust salt and pepper, if desired. Serve chicken with sauce.