While the noodles are cooking, place all remaining ingredients in a blender or food processor and pulse until smooth. Pour sauce into a large bowl.
When the noodles are done, drain, rinse well under cold water and drain again. Place udon noodles in the bowl and toss to evenly coat with sauce. Cover and refrigerate until chilled, 1 to 2 hours.
Toss again before serving. Serve with cooked spinach on the side and sprinkled sesame seeds and scallions on top.