1. Bring a large pot of cold water to a boil over high heat. Generously salt boiling water, add pasta, and cook, stirring occasionally, until al dente, about 8 to 10 minutes. Drain and rinse under cold running water, transfer to a large bowl and toss with sesame oil.
2. Chop garlic and ginger in a blender. With motor running, add peanut butter, soy sauce, brown sugar, vinegar, and red pepper and blend until smooth. Blend in 1/4 cup water. Pour over reserved spaghetti and toss to distribute.
3. Top pasta with cucumber, chicken, scallions, and peanuts to serve.
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