Heat 1/4 cup olive oil in a large skillet and cook and stir shallot and garlic until shallot is translucent, about 5 minutes. Stir 1 3/4 tablespoon Parmesan cheese, pine nuts, and basil into the hot olive oil, cooking until basil is tender and wilted, about 10 more minutes.
Stir 1 cup cream into basil mixture and bring to a simmer. Reduce heat to low and simmer sauce until it has reduced by 1/3, about 15 minutes. Stir remaining 1/2 cup cream into sauce and simmer until thickened again, 15 to 30 more minutes. Stir occasionally.
Heat 1 tablespoon olive oil in a separate skillet over medium heat and cook chicken until lightly browned outside and no longer pink inside, 10 to 12 minutes.
Bring a large pot of lightly salted water to a boil. Cook the bow-tie pasta at a boil, stirring occasionally, until cooked through yet firm to the bite, about 12 minutes; drain. Gently fold cooked pasta and chicken into basil sauce. Sprinkle with 1 tablespoon Parmesan cheese.
