1. Bring a large pot of water to a boil. Generously salt and cook the pasta according to the package's directions for al dente. Drain, shaking occasionally to prevent sticking, until cooled to room temperature.
2. Meanwhile, zest the limes into a large bowl and squeeze in 1/3 cup juice. Whisk in the fish sauce, brown sugar, vinegar, 1 tablespoon oil, and a pinch each salt and pepper.
3. Heat the remaining tablespoon oil a large skillet over medium-high heat. Season the shrimp with salt and pepper. Add to the skillet along with half the garlic. Cook, stirring occasionally, just until the shrimp is cooked through, about 3 minutes. Transfer to the bowl with the dressing.
4. Season the chicken with salt and pepper. Add to the same skillet along with the remaining garlic. Cook, stirring occasionally, until the chicken is browned and cooked through, about 5 minutes. Add the bean sprouts and scallion greens and cook, stirring, until barely wilted, about 1 minute. Transfer to the bowl with the dressing.
5. Add the carrots, pasta, and peanuts to the bowl. Toss until everything is well-coated. Season to taste with salt and pepper. Garnish with the scallions and remaining peanuts. Serve warm, at room temperature, or refrigerate and serve cold.