In a medium-sized bowl, whisk together the brown sugar, sesame oil, soy sauce, tahini, White Purée, vinegar, and garlic. Add red pepper flakes gradually to taste.
Put the pasta in a large serving bowl. Spoon the sauce over the pasta, tossing to coat the pasta evenly. Garnish with sliced green onions, cucumber, and sesame seeds, if using. Refrigerate until cool, or feel free to serve it warm.