Heat the olive oil in a 12-inch skillet over medium-high heat. Once the oil is glistening, add the sausage and cook until browned on both sides, 4 to 5 minutes.

Add the garlic and cook until fragrant, about 30 seconds. Add two thirds of the beans and 1 cup of the chicken stock. Season with the salt, pepper, and thyme. Stir to combine and bring to a simmer over medium heat.

In a medium bowl, mash the remaining beans with a fork until a thick paste forms. Add the smashed beans to the skillet and stir to combine. Simmer until the beans are heated through and the stock is bubbling, 2 to 3 minutes.

Add the kale, reduce the heat to medium-low, and cook, stirring often, until the sauce becomes creamy, is slightly reduced, and the kale has wilted, about 2 more minutes.

Stir in the Parmesan, lemon juice, and sausage. Add the remaining stock a little at a time if you need more creaminess. Adjust the seasoning to taste.

Serve with extra Parmesan and red pepper flakes, if desired, and crusty bread alongside.