Bring water to a boil in a large pot. Add soba noodles; cook, stirring occasionally until tender, about 5 minutes. Drain and rinse with cold water. Set aside or refrigerate until ready to eat.
Whisk together peanut butter, water, soy sauce, lime juice, and sriracha sauce until smooth.
Toss cooked soba noodles, carrots, celery, and green onions together in the peanut butter sauce. Garnish with cucumber and toasted sesame seeds. Serve cold.
