Bring a large pot of lightly salted water to a boil. Cook shirataki noodles in the boiling water, stirring occasionally, until tender yet firm to the bite, about 3 minutes. Drain and rinse again.
Heat olive oil in a skillet over medium heat. Add chicken, mushrooms, zucchini, sweet red pepper, and basil. Cook and stir until vegetables are mostly tender, about 5 minutes. Add cooked noodles, onion, and chile paste. Cook and stir until onion is translucent and vegetables are softened to your liking, about 5 minutes more. Sprinkle with grated Parmesan cheese.