Combine the tahini, peanut butter, soy sauce, vinegar, sugar, garlic, ginger, sesame oil and sambal oelek in the bowl of a food processor. Process to form a thick paste. With the motor running, slowly add the water (as needed) through the feed tube and process to form a thick dressing. Transfer to a bowl or container with a pour spout, and cover.
Bring a large pot of salted water to a rolling boil over high heat. Add the noodles and cook according to the "al dente" package directions. Drain and rinse with cold water, then transfer the drained noodles to a large bowl and pour about one-third of the dressing over them; toss to coat thoroughly.
Add the cucumber, scallions, carrot and cooked chicken, then pour another one-third of the dressing on top. Toss gently but thoroughly until everything is well coated. Add the remaining dressing if needed; the noodles and other ingredients should be well coated, but not drowning, in the dressing.
Cover and refrigerate for at least 2 hours, until the noodles are thoroughly chilled. Let sit at room temperature for about 10 minutes before serving.
