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1
For chicken, dissolve salt in 2 quarts cold tap water in a large bowl: submerge chicken in brine and refrigerate until fully seasoned, about 30 minutes.
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2
Rinse chicken pieces under running water and pat dry with paper towels.
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3
Season with pepper.
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4
Adjust oven rack to lowest position and heat oven to 450u00b0F.
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5
Heat oil in heavy-bottomed 12-inch ovenproof Skillet over medium-high heat until begining to smoke; swirl skillet to coat with oil.
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6
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.
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7
Turn chicken skin-side down and place in oven.
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8
Roast until juices run clear when chicken is cut with a paring knife, or thickest part registers 160u00b0F on instant-read thermometer, 15 to 18 minutes.
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9
Transfer chicken to platter, and let rest while making sauce.
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10
SAUCE: 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 1/2 minutes.
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11
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 3/4 cup, about 5 minutes.
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12
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.
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13
Spoon sauce around chicken breast and serve immediatly.
