Step 1: Adjust an oven rack to the lowest position and heat the oven to 450°F Pat the chicken dry with paper towels and season with salt and pepper. Heat the oil in a 12" oven-safe skilled over medium-high heat until just smoking. Carefully lay the chicken breasts skin-side down in the skillet and cook until well browned, 6-8 minutes. Turn the chicken breasts and continue to brown lightly on the second side, about 3 minutes.
Step 2: Turn the chicken beasts skin-side down and transfer the skillet to the oven. Roast until the thickest part of the breasts registers 160F on an instant read thermometer - about 15-20 minutes.
Step 3: Remove the skillet from the oven (warning: The handles are HOT!). Transfer the chicken breasts to a plate, tent loosely with foil, and let rest while making the sauce.
Step 4: Sauce: Being careful of the hot pan handle, add the oil to the skillet and return to medium heat until shimmering. Add the shallot and cook until softened, about 2 minutes. Stir in the flour and cook for 30 minutes. Stir in the broth, scraping up any browned bits. Bring to a simmer and cook 2-3 minutes to reduce slightly.
Step 5: Stir in any accumulated juices from the chicken and return to a simmer. Remove from heat and add remaining ingredients. Add salt and pepper to taste.
Step 6: Carve chicken and serve with sauce.
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