Brown the Chicken: Heat the oil in a wide skillet over medium-high. Place the chicken breasts in the hot pan and brown well on each side (it doesn’t have to be cooked through yet, just nicely browned). Remove from the pan and cover on a plate.
Sauté the Aromatics: Bring the pan back up to medium-high heat (do not drain or wipe the pan! That’s delicious flavor in there!) and add the bacon. Cook for a minute, then stir in the onion and garlic. Sauté until softened, about 1 minute.
Deglaze the Pan: Pour the chicken broth into the pan and bring to a boil, scratching the browned bits off the bottom. Stir in both mustards and season with salt and pepper.
Cook the Chicken: Add both the rosemary sprigs and the chicken to the pan with the sauce and cover with a lid. Gently simmer for about 6-8 minutes (if you have very thick chicken pieces, it might take longer), turning the chicken once halfway through. Once the internal temperature of the thickest part of the chicken registers 165°F, remove it from the pan onto a clean serving platter and cover.
Finish the Sauce: Bring the sauce to a boil, then cook until reduced by 1/2. Remove the rosemary and stir in the honey. Serve over the chicken.