Give the chicken thighs a liberal amount of salt and pepper. In a big skillet set over medium-high heat, warm up the olive oil. Place the chicken thighs skin side down and cook for approximately 5 minutes on each side, or until golden brown.
After taking the chicken out of the skillet, set it aside. Add the garlic and onion slices to the same skillet. Simmer for three minutes, or until tender. Stir in the chicken broth and apple cider, making sure to scrape up any browned bits from the skillet's bottom.
The chicken should be cooked through and tender after 30 to 40 minutes of braising. Take out of the skillet and reserve the chicken for later use. Turn up the heat to medium-high and let the sauce boil until it thickens a little.
Add the butter and stir until it melts, then taste and add salt and pepper to the sauce. Spoon the sauce over the chicken thighs and, if preferred, sprinkle with fresh sage leaves.