Add 2 quarts hot water to a large pan or large bowl.
Whisk in salt and brown sugar until dissolved.
Add 2 quarts cold water and hot sauce. Squeeze lemon into brine and add lemon rinds, rosemary, garlic and peppercorns.
Lower chicken into brine, totally submerging in brine. Cover and refrigerate at least one hour and up to 6 hours.
Drain chicken in a colander and discard brine solution.
Prepare a grill fire to 350° with hickory or pecan for smoke flavor.
Pat chicken dry and place on a baking sheet.
Rub butter under the skin and in the cavity of the chicken.
Sprinkle dry rub over the chicken on all sides.
Truss or tie up chicken legs and wings.
Grill chicken over indirect fire with the grill closed for one hour. Use a digital meat thermometer to check chicken in the thickest part of the thigh, near the bone. Cook until chicken reaches 165°.
Remove to a serving platter and rest 10 - 15 minutes before serving.