Bring a pot of lightly salted water to a boil and put the rigatoni into it. Cook until al dente (tender firm).
Heat a large skillet until very hot and then generously drizzle in some olive oil. Swirl to coat the skillet, and then add half of the cut-up chicken pieces, spreading them out as you put them in. NOTE: Do not begin stirring the chicken immediately as you want to get them nice and brown. After a minute or two, flip over the chicken with a spatula and let brown on the other side. Remove to a plate and set aside. Repeat with the second half of the chicken pieces.
Add a tablespoon or so of olive oil, if needed, into the hot skillet and throw in the garlic and chopped onions and give them a stir. Now add the chicken broth (or wine), whisking to deglaze the bottom of the skillet. Cook until the liquid reduces by half. Add the crushed tomatoes and stir to combine. Add salt and pepper to taste and a pinch of sugar. Reduce the heat and simmer for 15 minutes.
Add the chicken (and don't forget all of those yummy chicken juices on the plate) to the tomato/onion mixture and continue simmering for another 15 minutes. Toward the end of the cooking process, chop up your fresh basil and add it to the sauce, stirring to combine.
Put the pasta on a platter or in a large bowl and smother with the sauce. Top with grated Parmesan cheese.