In a 12-inch nonstick skillet over medium-high, heat the oil until shimmering. Add the onion, ¾ teaspoon salt and ½ teaspoon pepper. Cook, stirring occasionally, until the onion is soft and golden brown, about 10 minutes. Add the kale a large handful at a time, stirring to slightly wilt after each addition. Add ¼ cup water, then reduce to medium, cover and cook, stirring occasionally, until the kale is tender, 10 to 12 minutes. Transfer the kale mixture to a medium bowl; set aside. Wipe out the skillet.

Meanwhile, spread butter over 1 side of each slice of bread, evenly dividing the butter. Flip the slices to be buttered side down, then spread about ½ teaspoon mustard on each slice. Top each of 4 bread slices with 1 slice of cheese, a quarter of the kale mixture and 1 slice of bacon (if using), torn to fit. Top each with a slice of the remaining cheese, then with another slice of bread, buttered side up. Press on the sandwiches to compact the fillings.

Heat the skillet over medium until droplets of water flicked onto the surface quickly sizzle and evaporate. Add 2 of the sandwiches and cook until golden brown on the bottoms, 2 to 3 minutes. Using a wide spatula, flip the sandwiches and cook, pressing down lightly and adjusting the heat as needed, until golden brown on the second sides and the cheese is melted, 2 to 3 minutes. Transfer to a cutting board. Cook the remaining sandwiches in the same way (the second batch may cook faster). Cut each sandwich in half.