Prepare the chicken: Pat chicken breasts dry with paper towels and season generously on both sides with salt and pepper. If breasts are very thick, pound them gently to even thickness (about ¾ inch) for more even cooking.
Sear the chicken: Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Once hot, add chicken breasts and cook undisturbed for 5-6 minutes until golden brown. Flip and cook for another 5-6 minutes until internal temperature reaches 165°F. Transfer to a plate and tent with foil.
Create the flavor base: In the same skillet, reduce heat to medium and add remaining tablespoon of butter. Add diced shallot and cook for 2-3 minutes until softened. Add garlic and cook for another 30 seconds until fragrant, being careful not to burn it.
Make the cider glaze: Pour in apple cider and apple cider vinegar, scraping the bottom of the pan to loosen any browned bits (that’s where the flavor is!). Bring to a simmer and cook for 5-7 minutes until reduced by half.
Finish the sauce: Whisk in Dijon mustard, thyme, rosemary, and maple syrup if using. Simmer for another 2-3 minutes until the sauce thickens slightly and coats the back of a spoon.
Return chicken to pan: Lower heat and return chicken breasts to the skillet, including any accumulated juices. Spoon the glaze over the chicken and simmer for 1-2 minutes until chicken is warmed through and well-coated with the glaze.
Serve immediately: Transfer chicken to plates and spoon additional glaze over the top. Garnish with additional fresh herbs if desired.