Heat sun-dried-tomato oil and butter in a large Dutch oven over medium heat until melted and shimmering. Add shallot; cook, stirring often, until softened, 2 to 3 minutes.
Stir in sun-dried tomatoes, tomato paste and garlic; cook, stirring constantly, until the tomatoes are slightly broken down and deep red in color, about 2 minutes.
Stir in cooked shredded chicken, chicken broth, kale and salt. Bring to a boil over medium heat, cook, uncovered and stirring occasionally. Until the kale is tender and the mixture is slightly reduced, 15 to 20 minutes.
Remove from heat. Add 6 tablespoons Parmesan, heavy cream and [optional] cream cheese; stir until melted and combined, about 1 minute, Stir in basil and the remaining 2 tablespoons of Parmesan. Garnish with basil leaves, if desired.
