Place chicken in a medium bowl. Process garlic, cilantro roots, oyster sauce, soy sauce, brown sugar, fish sauce, pepper, and salt in a small food processor until smooth. Using a rubber spatula, scrape marinade into bowl with chicken and mix well. Cover and chill at least 4 hours and up to 12 hours.

Place chiles, vinegar, brown sugar, and fish sauce in a small glass jar; cover and shake to combine. Do ahead: Dressing can be made 1 day ahead. Store at room temperature.

Cook garlic and oil in a small skillet over medium-low heat, stirring often, until garlic begins to turn light brown around the edges, about 2 minutes. Continue to cook, stirring constantly, until garlic is medium brown and crisp, about 1 minute more. Immediately strain oil through a fine-mesh sieve into a small heatproof bowl. Let cool. Do ahead: Garlic can be cooked 1 day ahead. Transfer garlic oil and crispy garlic to separate airtight containers and store at room temperature.

Cook noodles in a large pot of boiling water according to package directions. Drain and return noodles to pot. Pour in lukewarm water to cover and stir with a wooden spoon or rubber spatula to release as much starch from noodles as possible. Drain and repeat until water is clear. Drain well, shaking off excess moisture. Transfer noodles to a large bowl, pour in garlic oil, and toss to coat.

Prepare a grill for high heat; lightly oil grate. Grill chicken until lightly charred on both sides and an instant read-thermometer inserted into the thickest part registers 165°, 8–10 minutes. Let rest 5 minutes before slicing against the grain.

Add dressing to noodles and toss well to coat. Taste and add more fish sauce and/or vinegar if needed; noodles should be salty and sour with sweetness trailing behind. Add scallions, lettuce, cilantro, salt, half of peanuts, and half of crispy garlic and toss to combine.

Transfer noodles to a platter and arrange chicken on top. Sprinkle crispy garlic and remaining peanuts over. Do ahead: Noodles can be cooked and dressed with oil 6 hours ahead. Store tightly covered at room temperature.