Pulse ginger, garlic, and sugar together in a food processor or blender. Add water, tamari, almond butter, 2 tablespoons sesame oil, Worcestershire sauce, rice vinegar, and red pepper flakes. Process again until smooth. Transfer sauce to a small bowl. Cover with plastic wrap and refrigerate until flavors intensify, at least 1 hour.
Bring a large pot of lightly salted water to a boil. Cook noodles in the boiling water, stirring occasionally, until tender yet firm to the bite, about 14 minutes. Toss in bok choy; cook until slightly wilted, about 1 minute. Drain and rinse in cold water to chill the noodles. Toss in a bowl with remaining 1 tablespoon sesame oil.
Heat a skillet over medium heat. Add carrots; cook and stir until slightly tender, about 3 minutes. Add zucchini. Cook until tender but not mushy, about 5 minutes.
Mix the chilled sauce into the bowl of noodles. Add the carrots and zucchini mixture, scallions, and sesame seeds.
