Dissolve salt in 2 quarts cold tap water in large container or bowl; submerge chicken in brine and refrigerate until fully seasoned, about 30 minutes. Rinse chicken pieces under running water and pat dry with paper towels. Season chicken with pepper.

Adjust oven rack to lowest position and heat oven to 450 degrees.

Heat oil in heavy-bottomed 12-inch ovenproof skillet over medium-high heat until beginning to smoke; swirl skillet to coat with oil. Brown chicken skin-side down until deep golden, about 5 minutes; turn chicken pieces and brown until golden on second side, about 3 minutes longer. Turn chicken skin-side down and place skillet in oven. Roast until juices run clear when chicken is cut with paring knife, or thickest part of breast registers 160 degrees on instant-read thermometer, 15 to 18 minutes. Transfer chicken to platter, and let rest while making sauce. (If not making sauce, let chicken rest 5 minutes before serving.)

Using potholder to protect hands from hot skillet handle, pour off most of fat from skillet; add shallot, then set skillet over medium-high heat and cook, stirring frequently, until shallot is softened, about 1 ½ minutes. Add chicken broth, vermouth, and sage; increase heat to high and simmer rapidly, scraping skillet bottom with wooden spoon to loosen browned bits, until slightly thickened and reduced to about ¾ cup, about 5 minutes. Pour accumulated chicken juices into skillet, reduce heat to medium, and whisk in butter 1 piece at a time; season to taste with salt and pepper and discard sage. Spoon sauce around chicken breasts and serve immediately.