Pat 4 chicken cutlets dry with paper towels. Sprinkle both sides with ¾ teaspoon pepper and ¼ teaspoon salt.

Heat 2 tablespoons oil in a large skillet over medium-high heat. Add the chicken and cook until golden, 2 to 3 minutes per side. Transfer to a plate and cover to keep warm. Do not wipe the skillet clean.

Add sliced onion to the skillet; cook over medium heat, stirring occasionally, until starting to brown, about 4 minutes.

Add ¼ cup broth; cook, stirring constantly and scraping up browned bits, until the liquid has evaporated and the onion is softened, about 2 minutes.

Add apple slices; cook, stirring often, until beginning to soften, 3 to 5 minutes.

Stir in 1 tablespoon mustard and the remaining ¾ cup broth and ¼ teaspoon salt; bring to a boil over high heat.

Return the chicken to the skillet, nestling it into the onion mixture. Reduce heat to maintain a gentle simmer; cook until the sauce has slightly reduced and an instant-read thermometer inserted into the thickest part of the chicken registers 155°F, about 5 minutes.

Stir in ¼ cup cream and return to a simmer. Reduce heat to maintain a gentle simmer; cook, stirring occasionally and flipping the chicken once, until the sauce thickens slightly and the chicken reaches 160°F, about 7 minutes more.

Remove from heat and stir in 1 tablespoon butter until melted. (The chicken will continue cooking to 165°F from the residual heat.) Garnish with thyme leaves and/or additional pepper, if desired.