In a large pot of boiling salted water, cook pasta until al dente, according to package instructions. Drain, and transfer to a large bowl.
Meanwhile, make peanut sauce: In a blender or food processor, blend peanut butter, garlic, pepper flakes, soy sauce, vinegar, and sesame oil until smooth.
Cut cucumber crosswise into thirds; cut each piece lengthwise into 1/4-inch slices. Stack slices; cut lengthwise into very thin strips. Transfer pasta to a large bowl; toss with sauce, cucumber, and carrots. Serve at room temperature.