In a food processor, pulse garlic and ginger until finely chopped. Add chili paste, peanut butter, soy sauce, sugar, peanut oil, lime juice, and 4 tablespoons water; puree until smooth. Set aside.
Bring a large stockpot of salted water to a boil. Add noodles; cook until al dente, following label directions. Drain in a colander; rinse with cold water. In a large bowl, toss noodles with 1 cup peanut sauce. Set aside.
Cut cucumbers into 36 (3/4-inch) rounds; scoop out the seeds with a melon baller, forming deep cups. Dab some peanut sauce on the bottom of a cucumber cup. Using a small cocktail fork, twirl a few noodles together, and place in cup. Garnish with chopped peanuts and scallions. Repeat with remaining cucumber cups.
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