Combine first 10 ingredients in a small saucepan; stir with a whisk until blended. Bring to a boil; cook 1 minute, stirring constantly. Set aside.
Cook noodles in boiling water 8 minutes. Drain well. Combine noodles, peanut sauce, bok choy, snow peas, and carrot in a large bowl, and toss well to coat.
Note: Chile paste with garlic can be found with other ethnic foods in the supermarket. It can be omitted if you perfer a not-so-hot sauce. If udon noodles are difficult to find, try an Asian market.