Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until cooked through but firm to the bite, 10 to 12 minutes. Drain and rinse under cold water. Transfer pasta to a serving bowl and toss with olive oil.

Whisk soy sauce, sugar, vinegar, sesame seeds, chili sauce, and sesame oil together in a bowl until sugar dissolves. Toss soy sauce mixture with pasta; top with green onions, red bell pepper, and snap peas. Refrigerate 30 minutes to overnight to allow flavors to blend. Toss again before serving.