Place chicken, pickle juice, and brown sugar in a large ziptop bag. Seal bag and place it in a bowl. Refrigerate for 24 hours.
Remove chicken from brine. Pat dry with paper towels.
Season with salt, pepper, and Cajun seasoning.
Grill over medium-high heat for 5 to 6 minutes per side.
Notes
Boneless, skinless chicken breasts can be used instead of thighs. They will cook a little more quickly. Chicken is done when it reaches an internal temperature of 165 degrees.