For cowboy butter, stir together butter, garlic, parsley, chives, mustard, lemon zest, smoked paprika, lemon juice, thyme, chili powder, and crushed red pepper in a medium bowl. Wrap in plastic wrap and chill 2 hours, or until ready to use.
Cut chicken in bite-size pieces. Toss chicken to coat with oil, salt and pepper in a medium bowl. Thread on eight 6-inch skewers leaving about 1/4-inch between pieces.
Place skewers on the oiled rack of an outdoor grill directly over medium heat. Cover and grill until no longer pink (165 degrees F or 74 degrees C), 8 to 10 minutes, turning once. Or preheat an air fryer to 400 degrees F (200 degrees C). Place skewers in the air fryer basket. Cook until no longer pink (165 degrees F or 74 degrees C), 11 to 12 minutes, turning once.
Melt half of the cowboy butter. Drizzle over the cooked chicken skewers to coat. Serve with remaining cowboy butter.