Cook noodles according to package directions. Drain noodles and rinse under cold water. Once cool, transfer noodles to a large mixing bowl and set aside.
Grate ginger to measure one tablespoon. Combine ginger, chili garlic sauce, vinegar, sesame oil, soy sauce, peanut butter, brown sugar and hot water and whisk until combined.
Slice green onions diagonally and cut cucumber into julienne strips. Cut carrots into long ribbons using a peeler and cut ribbons crosswise into 1-inch pieces. Add veggies to noodles and toss with dressing. Garnish with sesame seeds if desired. Serve cold or at room temperature.