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Preheat oven to 400 degrees.
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In a medium bowl, mix together vinegar, fish sauce, sambal paste, soy sauce, sesame oil, and lime juice. Slowly whisk in 1/4 cup vegetable oil to emulsify. Stir in 1 slice red chile and 1/4 cup chopped cilantro; set vinaigrette aside.
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Heat a grill pan over high heat. Rub chicken with 2 tablespoons oil and season with salt and pepper. Place chicken on grill pan and cook, turning once, about 2 minutes per side. Transfer chicken to a baking sheet; transfer to oven and bake until cooked through, about 10 minutes. Transfer to a cutting board and let cool.
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Fill a medium saucepan with oil to a depth of 3 inches; heat oil over medium-high heat until it reaches 375 degrees on a deep-fry thermometer. Working in batches, add cellophane noodles to hot oil and cook until puffed, about 30 seconds. Transfer to a paper-towel lined plate to cool.
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Bring a medium pot of water to a boil over high heat. Prepare an ice-water bath and set aside. Add snow peas to boiling water and cook until tender-crisp, about 2 minutes. Drain and transfer to ice-water bath until cooled. Drain and thinly slice on the bias.
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Arrange cabbage, carrots, cucumbers, noodles, and snow peas on a large platter. Slice cooled chicken and arrange over salad. Garnish with peanuts, basil, mint, and remaining slices of chile. Serve with vinaigrette.
