In a large heavy-bottomed pan with a tight-fitting lid, cook sausage over medium-high heat, stirring occasionally, until browned, 4 to 6 minutes; using a slotted spoon, transfer to a paper-towel-lined plate to drain. Set aside.
Add kale to pan in two or three batches, waiting for one batch to wilt before adding the next. Cook over medium until bright green, about 1 minute. Add 1/2 cup water. Reduce heat to medium-low; cover, and simmer until tender, 15 to 20 minutes, stirring once or twice. Uncover; cook until most of the liquid has evaporated. Stir in vinegar; season with salt and pepper. Serve topped with sausage.