In a large pot of boiling, salted water, cook pasta, stirring occasionally, until al dente, about 12 minutes.
Meanwhile, in a medium skillet, melt butter in vegetable oil. Season chicken with salt and pepper and cook over medium-high heat until lightly browned and firm to touch, about 5 minutes per side. Transfer to a plate and cover with foil to keep warm. Discard all but 1 Tbsp. fat from skillet. Add tomatoes and cook over medium-high heat, stirring often, until they start to soften, about 5 minutes. Cut chicken into chunks.
Ladle off 1 cup pasta water and set aside. Drain pasta and return to pot. Add chicken, tomatoes and reserved pasta water. Stir in pesto and mix well until ingredients are coated.