To make the marinade: Whisk coconut milk, garlic, brown sugar, curry powder, salt, and pepper together in a large glass or ceramic bowl. Add chicken and toss to evenly coat. Cover the bowl with plastic wrap and marinate in the refrigerator for at least 2 hours.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Remove chicken from marinade and shake off excess. Thread marinated chicken onto skewers; discard remaining marinade.
To make the peanut sauce: Whisk coconut milk, chicken stock, peanut butter, brown sugar, and curry powder together in a saucepan over medium-high heat until simmering. Reduce the heat and simmer, whisking constantly, until smooth and thickened, about 5 minutes. Remove from the heat and stir in lime juice, soy sauce, and salt.
While the sauce is coming to a simmer, cook chicken skewers on the preheated grill until no longer pink in the center and the juices run clear, 4 to 5 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Serve skewers with warm peanut sauce.
