To brine chicken
In a 3 quart saucepan, stir together brine mix, 4 cups water and 2 lemon halves. Bring mixture to a boil. Reduce heat and simmer for 5 minutes. Remove pan from heat and cool; stir in ice.
Place chicken in a brining bag. Pour cooled brine mix over chicken and close bag. Refrigerate overnight.
The next day, remove chicken from refrigerator. Discard brine and rinse chicken well. Place 4 lemon halves inside chicken. Carefully lift chicken skin along breast and legs being careful not to tear it.
Rub 4 tablespoons garlic-rosemary butter under skin. Truss chicken with kitchen twine. Rub with oil and sprinkle with salt and pepper.
To make the garlic-rosemary butter, in the small bowl of a food processor, combine garlic, rosemary, salt, tarragon and butter. Reserve.
To make the potatoes, wash well and dry on a fresh, lint-free towel. Place potatoes in a 10" x 14" baking dish and drizzle with olive oil. Sprinkle with rosemary and salt.
Place chicken on rotisserie and mount in oven. Place baking dish filled with potatoes on bottom most rack in oven, directly under chicken. The juices from the chicken will help flavor the potatoes. Cook for 1 1/2 to 2 hours, or until internal temperature of the chicken reads 170 F.
Remove chicken from rotisserie. Discard string and lemons. Slice chicken and serve with the potatoes.