Lightly smash garlic cloves with the side of a large knife to loosen the skins. Peel the cloves, cutting large cloves in half.
Heat the olive oil in a cast iron (or other oven-safe) skillet over medium-high heat. Add the garlic and cook until it begins to brown, about two minutes. Remove from pan and set aside.
Season both sides of the chicken breasts with salt and pepper. Add chicken to the pan and cook until browned on one side, about 4 minutes. Turn the chicken.
Whisk together the chicken broth, wine, mustard, flour and remaining 1/4 teaspoon of salt until smooth.
Return the garlic cloves to the pan and pour in the wine mixture. Move pan to the oven and cook for 30 minutes, or until chicken is cooked through (internal temperature should be 165 degrees).
Whisk the sauce briefly to smooth out any lumps.
Serve the chicken sprinkled with chives or scallion greens. Drizzle sauce over the chicken. (The sauce is also excellent on potatoes.)