2 quarts water
3 1/2 cups cider vinegar
2 tbsp salt
2 (3 1/2-to 4-pound) whole chickens , halved (see tip)
Seasoning and Sauce
1 tablespoon ground poultry seasoning
Salt and pepper
1/2 cup cider vinegar
3 tablespoons Dijon mustard
1 tablespoon chopped fresh sage leaves
1 tablespoon chopped fresh rosemary
1/2 cup olive oil
For the chicken: Whisk water, vinegar, and salt in large bowl until salt dissolves. Add chicken and refrigerate, covered, for 1 to 2 hours. (This is called brining the chicken.)
For the seasoning and sauce: Combine poultry seasoning, 2 teaspoons salt, and 2 teaspoons pepper in small bowl; set aside. Process vinegar, mustard, sage, rosemary, 1/2 teaspoon salt, and 1/2 teaspoon pepper in blender until smooth, about 1 minute. With blender running, slowly add oil until incorporated. Transfer vinegar sauce to small bowl and reserve for basting chicken in step 4.
Heat all burners on high for 15 minutes, then turn all burners to medium-low. (For charcoal grill, light 75 coals; when covered with fine gray ash, spread evenly over bottom of grill. Set cooking grate in place and heat covered, with lid vents open completely, for 5 minutes.) Scrape and oil cooking grate.
Remove chicken from brine. Pat dry with paper towels and rub all over with poultry seasoning mixture. Arrange chicken skin-side up on grill and baste with vinegar sauce. Grill, covered, until chicken is well browned on bottom and meat registers 120 degrees, 25 to 30 minutes, basting with sauce halfway through cooking. Flip chicken skin-side down and baste with sauce. Continue to grill, covered, until skin is golden brown and crisp and thigh meat registers 170 to 175 degrees, 20 to 25 minutes longer. Transfer chicken to platter (do not cover) and let rest 5 minutes. Serve.
Make Ahead: Both the spice rub and the sauce can be made up to 3 days ahead. The chicken can be brined, patted dry, and rubbed with the seasoning mixture up to 8 hours ahead.