Bring a large pot of lightly salted water to a boil. Cook rigatoni in the boiling water, stirring occasionally until tender yet firm to the bite, about 13 minutes. Drain.
Heat butter and oil together in a large skillet over medium-high heat. Add garlic and shallot to the skillet; cook and stir until fragrant and shallots are translucent, 2 to 3 minutes. Add chicken; cook and stir until it is no longer pink and the internal temperature reads at least 165 degrees F (74 degrees C), about 5 minutes.
Mix cherry peppers into chicken mixture; cook until soft, 5 to 7 minutes. Stir in tomato sauce; mix thoroughly. Lower heat to medium and simmer, 7 to 10 minutes.
Pour sherry into tomato sauce mixture; simmer, stirring occasionally, about 10 minutes. Pour in milk and simmer, about 5 minutes.
Remove from heat and mix in torn basil leaves. Spoon tomato sauce over pasta to serve.