Preheat the oven to 400˚ F. Bring a medium pot of water to a boil. Stir the lime juice, fish sauce, soy sauce, sugar and ginger in a small bowl until the sugar is dissolved. Toss the chicken with 2 tablespoons of the sauce and the grated garlic; season with salt and pepper. Let marinate 10 minutes.
Heat 2 tablespoons vegetable oil in a large nonstick skillet over medium-high heat. Pat the chicken dry. Add to the skillet and cook until browned, 3 minutes per side. Transfer the chicken to a baking sheet and bake until cooked through, 8 to 10 minutes. (Reserve the skillet.) Let the chicken rest 5 minutes, then slice.
Meanwhile, add the noodles to the boiling water and cook as the label directs. Drain in a fine-mesh sieve and rinse under cold water. Shake off any excess water.
Heat the remaining 2 tablespoons vegetable oil in the reserved skillet over medium-high heat. Add the snow peas, chopped garlic and a big pinch of salt. Cook, stirring, until the snow peas are bright green, 1 minute. Add the noodles and stir until hot, 2 minutes. Season with salt and pepper; stir in half the cilantro.
Divide the noodles and chicken among plates. Drizzle with the remaining sauce and top with the remaining cilantro.