Zest limes, halve, and juice. Mince cilantro (no need to stem). Core apples and cut into ¼" slices. Stack slices, cut into matchsticks, and place in a large mixing bowl with water. Coarsely chop peanuts. Rinse chicken and pat dry. On a separate cutting board, slice each chicken breast lengthwise into thirds, then halve those pieces once more to create 24 long strips. Season both sides of strips with ¾ tsp. salt and ¼ tsp. pepper.
Marinate the Chicken
Combine half the honey (reserve remaining for slaw), paprika, and 1 ½ Tbsp. olive oil in a medium mixing bowl. Add chicken strips and toss to coat. Thread two pieces of chicken onto each skewer and set aside on a plate to marinate.
Make the Slaw
Combine remaining honey, 2 Tbsp. lime juice, 2 Tbsp. olive oil, ¼ tsp. salt, and a pinch of pepper in a small bowl. Drain water from mixing bowl with apples, add slaw mix, honey-lime juice dressing, and toss to combine. Set aside to let flavors marry.
Make the Cilantro-Lime Pesto
Combine cilantro, chopped peanuts, 2 tsp. lime zest, 1 tsp. lime juice, and 3 Tbsp. olive oil in a small mixing bowl.
Grill the Chicken
Heat a grill pan or outdoor grill over medium-high heat. Add chicken skewers to hot grill and cook until well-charred and chicken reaches a minimum internal temperature of 165 degrees, 3-4 minutes per side.
Plate the Dish
Place a serving of apple slaw on a plate with chicken skewers in front. Top skewers with cilantro-lime pesto or serve on side. Garnish with remaining lime zest.