Dissolve salt in 2 quarts cold water in large container. Submerge chicken pieces in brine, cover, and refrigerate for at least 30 minutes or up to 1 hour.
Adjust oven rack to middle position and heat oven to 450 degrees. Remove chicken from brine and pat dry with paper towels. Sprinkle chicken with pepper. Heat oil in 12-inch ovensafe skillet over medium-high heat until just smoking. Cook chicken skin side down until well browned, 6 to 8 minutes, reducing heat if skillet begins to scorch. Flip chicken and brown lightly on second side, about 3 minutes. Flip chicken skin side down, transfer skillet to oven, and roast until chicken registers 160 degrees, about 10 minutes.
Using potholder, remove skillet from oven. Transfer chicken to serving platter and let rest while making sauce.
Being careful of hot skillet handle, pour off all but 1 teaspoon fat left in skillet. Add sage and cook over medium-high heat until fragrant, about 30 seconds. Stir in broth and vermouth, scraping up any browned bits, and simmer until thickened and measures ⅔ cup, about 6 minutes. Stir in any accumulated chicken juices. Reduce heat to low and whisk in butter, one piece at a time. Off heat, season with salt and pepper to taste. Spoon sauce over chicken and serve.
