Pat the chicken dry and season all over with salt. Heat a large heavy skillet over high heat and add the olive oil. When the oil is hot, add the chicken skin-side down and cook, undisturbed, until the skin is browned and crisp, about 5 minutes.
Turn the chicken and add the apples, shallots and sage to the skillet. Reduce the heat to medium high and cook until the chicken is browned on the bottom, 4 to 5 minutes. Transfer the chicken to a plate and continue to cook the apples and shallots, stirring, until golden, about 2 more minutes.
Meanwhile, make the glaze: Mix the chicken broth, maple syrup, vinegar and 1/2 teaspoon salt in a small bowl. Add the mixture to the skillet with the apples and shallots and boil until reduced by about three-quarters, 2 to 3 minutes. Return the chicken to the skillet, turning to coat, until cooked through, about 2 more minutes.