Cut potatoes into ¼" wedges. Trim cucumber, halve lengthwise, then cut into matchsticks. Zest and halve lime. Quarter one half and juice remaining half. Stem cilantro and mince stems. Reserve leaves whole. Coarsely chop peanuts. Pat chicken thighs dry.
Make the Potatoes
Toss potatoes on prepared baking sheet with 2 tsp. olive oil. Spread into a single player and roast until browned and tender, 18-20 minutes. Toss with half the seasoning blend (reserve remaining for chicken) and cilantro stems. While potatoes roast, make slaw.
Make the Slaw
Toss carrots, cucumber, cilantro leaves, peanuts, and 1 Tbsp. lime juice, 2 tsp. olive oil, and ¼ tsp. salt in a mixing bowl. Set aside to let flavors marry.
Sear the Chicken
Place a medium pan over medium-high heat. Add 2 tsp. olive oil and chicken thighs to hot pan. Cook until browned and chicken reaches a minimum internal minimum temperature of 165 degrees, 3-4 minutes per side. Remove from burner, cover, and rest 3 minutes.
Shred Chicken and Warm Buns
Shred chicken in pan and add soy sauce, 1 tsp. lime zest, and remaining seasoning blend. Stir until chicken is coated. Place buns directly on oven rack and toast until bread is warmed, 1-2 minutes.
Finish the Dish
Place shredded chicken on top of bottom bun. Top with slaw and top bun. Place sandwich on a plate, add potatoes on the side, and garnish with lime wedges.