Bring first 3 ingredients to a boil in a small saucepan over medium-high heat. Boil, stirring occasionally, 1 to 2 minutes or until salt and sugar dissolve. Remove saucepan from heat; let stand 10 minutes. Pour mixture into a large bowl; stir in 1 cup cold water, 1/2 cup tequila, and next 4 ingredients. Submerge chicken in brine; cover and chill 6 to 12 hours.
Coat cold grill grate with cooking spray. Preheat grill to 350° to 400° (medium-high) heat. Remove chicken from bowl; discard brine. Pat chicken dry with paper towels, and brush skin with olive oil. Sprinkle chicken with kosher salt and freshly ground black pepper. Let stand at room temperature 20 minutes.
Grill chicken, covered with grill lid, 40 to 45 minutes or until done, turning occasionally. Transfer chicken to a platter. Grill limes, cut sides down, 1 to 2 minutes or until charred. Serve limes with chicken.
Brooke Parkhurst and James Briscione, thecoupleskitchen.com