Heat 1 tablespoon olive oil in a 12-inch skillet or frying pan over medium-high heat until shimmering. Add 12 to 16 ounces sliced smoked sausage in an even layer and cook until browned, 3 to 4 minutes per side. Using tongs or a slotted spoon, transfer to a bowl or plate.
Add 3 minced garlic cloves to the pan and cook until fragrant, about 30 seconds. Add 1 (about 15-ounce) can white beans (do not drain) and use a potato masher or fork to mash about half of the beans. Stir in the remaining 1 can white beans (do not drain), ½ cup low-sodium chicken broth, 2 teaspoons dried Italian seasoning, ½ teaspoon black pepper, and ¼ teaspoon kosher salt. Bring to a simmer and cook until thickened and reduced slightly, 3 to 5 minutes.
Stir in ½ bunch coarsely chopped kale leaves and 2 ounces finely grated Parmesan cheese until the kale is wilted and the cheese is melted, about 1 minute.
Return the sausage and any accumulated juices to the pan. Stir in 1 tablespoon lemon juice. Ladle into bowls and serve with more Parmesan cheese if desired.
