Season the chicken on all sides with salt and pepper. In a 12-inch skillet over medium-high, heat the oil and 1 tablespoon butter. When the butter begins to foam, add the chicken and cook until golden brown on the bottom, about 3 minutes. Flip, reduce to medium and cook until lightly browned on the second sides, 2 to 3 minutes. Using tongs, transfer to a large plate.

Return the skillet to medium. Add the onion and ½ teaspoon salt; cook, stirring often, until golden brown, 3 to 5 minutes. Add the apples and thyme, then cook, stirring occasionally and gently, until the apples are golden brown and beginning to soften, about 3 minutes. Add the vinegar and broth; bring to a simmer, scraping up any browned bits. Return the chicken to the skillet, nestling the breasts in the apple mixture, and add any accumulated juices. Cover and cook, turning the chicken two or three times, until the thickest part of the breasts reaches 160°F, 5 to 7 minutes.

Remove the pan from the heat. Using tongs, transfer the chicken to a cutting board and let rest. Meanwhile, bring the apple mixture to a simmer over medium-high, then add the remaining 2 tablespoons butter. Cook, stirring, until the butter is incorporated and the sauce is lightly thickened, about 2 minutes. Off heat, remove and discard the thyme sprigs, if used. Stir in the parsley, then taste and season with salt and pepper.

Using a slotted spoon, transfer the apples to a platter. Slice the chicken about ½ inch thick against the grain, place it on the apples and spoon on the sauce.