Bring a large pot of salted water to a boil. Cook the spaghetti according to the package directions. Reserve about 1 cup of the pasta cooking water and drain.
While the pasta is cooking, make the sauce.
Heat the olive oil in a large skillet over medium-high heat. Add the onion and red pepper and cook until they begin to soften, about 3 minutes. Add the garlic and cook, stirring, for 30 seconds. Add the chicken and cook, stirring occasionally, until the chicken is browned and cooked through, about 6-8 minutes depending on the size. Add the Worcestershire and sweet chili sauces. Cook, stirring, until heated through. Add the spaghetti to the pan and toss to coat. Add some of the pasta cooking water if the sauce is too thick. Garnish with basil and serve.