Prepare the chicken. Pat the chicken cutlets dry. Season both sides with 1/2 teaspoon of the salt and 1/4 teaspoon of the black pepper. Sprinkle the flour over the chicken and turn each cutlet until very lightly coated, shaking off any excess.

Brown the apples. Melt the butter in a large 12-inch skillet over medium-high heat. Add the apple slices in an even layer and cook for 4 to 5 minutes, turning once, until lightly golden at the edges and just beginning to soften. Transfer the apples to a plate.

Sear the chicken. Add the olive oil to the skillet. Add the chicken cutlets and sear for about 3 minutes per side until golden. They do not need to be fully cooked at this stage. Transfer the chicken to a clean plate.

Build the cider sauce. Reduce the heat to medium. Add the onion, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon black pepper. Cook for 3 to 4 minutes until the onion begins to soften. Add the garlic and thyme and cook for about 30 seconds. Pour in the apple cider and chicken broth, then whisk in the Dijon mustard. Scrape up the browned bits from the skillet and simmer uncovered for 5 to 6 minutes, until reduced by roughly one-third.

Add the cream and finish cooking the chicken. Stir in the heavy cream and simmer for about 2 minutes. Return the chicken and any accumulated juices to the skillet. Reduce the heat to medium-low and simmer for 4 to 6 minutes, until the sauce lightly coats a spoon and the thickest part of each chicken cutlet reaches 165°F / 74°C.

Finish the skillet. Stir the apple cider vinegar into the sauce. Return the browned apples to the skillet and cook for about 1 minute, just until warmed through. Serve each chicken cutlet with apples and cider cream sauce spooned over the top.