Prepare grill for direct/indirect (or two-zone cooking). Target 350 degrees Fahrenheit in the cooking chamber.
Place 1 teaspoon of Cajun seasoning in the butter and mix. Using your fingers, slowly separate the skin from the meat on each breast to make a pocket. Add the butter into the pocket and press down to make smooth. Finish seasoning the rest of the breast with the 1/4 cup of Cajun rub.
Place the seasoned turkey breast over the direct heat skin side down. Grill 6 – 8 minutes or until the skin shows a little browning. Flip and grill the other side of the turkey for 4 – 6 minutes. Once a little browning occurs, move to indirect side of heat.
Continue grilling turkey breast over indirect heat until the internal temperature of the thickest part of the breast reads between 160 to 165 degrees Fahrenheit using an instant read thermometer. This should take between 30 – 45 minutes depending on the size of the breast.
Remove the breast from the grill, and let rest for 10 minutes. Slice and serve with your favorite sides.