Pat chicken pieces dry with paper towels and season generously with salt and pepper on all sides. Heat olive oil in your largest skillet over medium-high heat until shimmering, about 2 minutes.
Working in batches, sear chicken pieces until golden brown on both sides, approximately 4-5 minutes per side. Don't overcrowd the pan—this ensures proper browning and develops deep flavor.
Remove chicken from the skillet and set aside on a clean plate. Add diced onion to the remaining oil and cook for 3 minutes until softened, then add minced garlic and stir for 30 seconds until fragrant.
Pour in chicken broth and apple cider, scraping the pan's bottom to release flavorful browned bits. Return the cozy fall skillet chicken apple recipe to the pot and add sliced apples, thyme, sage, cinnamon, and nutmeg.
Bring the mixture to a gentle simmer, then reduce heat to medium-low and cover partially with a lid. Simmer for 15-18 minutes until the apples are tender and the chicken reaches an internal temperature of 165°F.
In a small bowl, whisk cornstarch with water to create a slurry. Stir this mixture into the simmering skillet and cook uncovered for 2-3 minutes until the sauce thickens slightly.
Remove from heat and swirl in butter until melted throughout, creating a silky sauce. Taste and adjust seasoning with additional salt, pepper, or spices as desired.