Spread the flour in a shallow dish. Pound the thicker end of each chicken breast to a uniform ½-inch thickness.

Pat the chicken dry with paper towels, then sprinkle with ⅛ teaspoon salt and ⅛ teaspoon pepper.

Working one at a time, dredge the chicken in the flour, shaking off any excess.

Melt 2 tablespoons butter in a 12-inch nonstick skillet over medium heat.

Add the apples, cut sides down, and cook until browned, about 5 minutes per side, moving and redistributing as needed.

Transfer to a plate and set aside.

Melt the remaining 1 tablespoon butter in the now-empty skillet over medium heat.

Add the chicken and cook until lightly browned, about 2 minutes per side.

Transfer the chicken to a separate plate.

Add the onion, thyme, ¼ teaspoon salt, and ¼ teaspoon pepper to the skillet. Cook over medium heat until softened and lightly browned, about 5 to 7 minutes.

Stir in the cider and broth, scraping up any browned bits. Bring the mixture to a simmer.

Return the chicken and accumulated juices to the skillet. Reduce the heat to medium-low, cover, and cook until the chicken registers 160°F, about 8 to 12 minutes.

Transfer the chicken to a serving platter and tent with foil.

Stir the cream and apples into the skillet. Increase the heat to medium-high and bring to a boil. Cook until the sauce thickens slightly, about 2 minutes.

Stir in the vinegar and accumulated chicken juices. Adjust the seasoning with additional salt and pepper as needed.

Spoon the sauce and apples over the chicken and sprinkle with chives. Serve immediately.