Heat oil in a skillet over medium-high heat. Add mushrooms, snow peas, carrot, and green onions. Cook, stirring, until crisp-tender, about 4 minutes. Add garlic; cook 1 minute more.

Stir together soy sauce, sugar, and cayenne in a small bowl; pour into skillet. Add cooked spaghetti and toss to coat. Cook until heated through, about 1 minute. Sprinkle with toasted sesame seeds.