1. Starting on thick side of breast, closest to breastbone, cut horizontal pocket in each breast, stopping 1/2 inch from edge so halves remain attached. Dissolve 3 tablespoons salt in 1 quart cold water in large container. Add breasts, cover with plastic wrap, and refrigerate 30 minutes.

2. Turn all burners to high, close lid, and heat until hot, about 15 minutes. Scrape grate clean with grill brush. Dip wad of paper towels in vegetable oil; holding wad with tongs, wipe cooking grate. Grill is ready when coals are hot (you can hold your hand 5 inches above grate for 2 seconds). Leave primary burner (burner that will stay on during grilling) on high and turn off other burners.

3. Meanwhile, combine butter, shallot, and tarragon in small bowl. Roll each piece of cheese in 1 slice prosciutto. Remove breasts from brine, dry thoroughly inside and out with paper towels, and season inside and out with pepper. Spread equal amount of butter mixture inside each breast. Place 1 prosciutto-wrapped piece of cheese inside each breast and fold breast over to enclose. Evenly space 3 pieces kitchen twine (each about 12 inches long) beneath each breast and tie, trimming any excess.

4. Place chicken breasts, skin side down, over hot part of grill and cook until well browned, 3 to 5 minutes. Flip chicken and cook until second side is just opaque, about 2 minutes. Move chicken, skin side up, to cool side of grill, with thicker side of breasts facing fire. Cover grill and continue to cook until instant-read thermometer inserted into thickest part of stuffing registers 160°F, 30 to 40 minutes. Transfer chicken to carving board and let rest, tented with foil, 10 minutes. Remove twine, then carve breast meat from bone (see photo below). Carve into 1/2-inch-thick slices and serve.

Serves 4