Bring 4 quarts water to boil over high heat. Meanwhile, whisk garlic, ginger, soy sauce, vinegar, hot sauce, brown sugar, and oil in small bowl. Toss chicken with 3 tablespoons garlic mixture in bowl. Puree remaining garlic mixture and peanut butter in blender until smooth.
Add 1 tablespoon salt and noodles to boiling water and cook until al dente. Drain well, rinse under cold running water until cool, and drain again. Toss noodles, peanut sauce, and scallions in large bowl. Adjust seasonings.
Grill chicken breasts over high heat until cooked through, 2 to 3 minutes per side. Slice chicken and serve with noodles.