Whisk the water, lime juice, sugar, and fish sauce in a small bowl until the sugar dissolves. Stir in the sambal oelek and garlic.
Cut the chicken thighs into 2-inch cubes. Toss cubes in a medium bowl with about 1/4 cup of the sauce and 1 teaspoon vegetable oil. Set aside for 15 minutes. Stir a heaping tablespoon of grated carrot into reserved sauce. Set aside.
Heat a grill pan or outdoor grill to medium-high heat. Thread 2 to 3 pieces of chicken on each skewer and lay skewers on a baking sheet or large plate. Season with salt and pepper. Working in batches, grill the satays, without moving them, until browned on the first side, about 3 minutes. Turn the satays and grill until the second side is browned and the meat is opaque at the edges, about 3 minutes more. Transfer to a large clean platter.
Arrange the lettuce, mint, cilantro, cucumber, scallions, and the rest of the carrot on the platter with the satays. To serve, slip the chicken from a skewer and wrap each piece in a lettuce leaf with some of the herbs and vegetables and dip in the sauce
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