Heat 1 tablespoon sun-dried tomato oil and 1 tablespoon butter in a large Dutch oven over medium heat until melted and shimmering. Add chopped shallot; cook, stirring often, until softened, 2 to 3 minutes.
Stir in ½ cup sun-dried tomatoes, 2 tablespoons tomato paste and the minced garlic; cook, stirring constantly, until the tomatoes are slightly broken down and deep red in color, about 2 minutes.
Stir in rinsed chickpeas, 6 cups broth, chopped kale and ½ teaspoon 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, ⅓ cup cream and the cubed cream cheese; stir until melted and combined, about 1 minute. Stir in 2 tablespoons basil. Divide among 4 bowls; top with the remaining 2 tablespoons Parmesan. Garnish with basil leaves, if desired.
