Heat oil in a large cast iron pan over medium heat. Add sausage and cook until browned. Remove sausage to a plate.
In the oil remaining in the pan, cook onion and garlic until translucent, about 4 minutes. Season with salt and pepper. Add red pepper flakes, cumin, and as much kale as will fit. Cook until slightly wilted and there is enough room to add more kale. Repeat until all of kale is in pan (for smaller pans, you may have to cook in batches). Add stock and cider or beer, reduce heat, and cover. Cook, stirring occasionally, until tender, about 30 minutes. Squeeze lemon juice over top. Add cooked sausage back to skillet, and stir to incorporate. Serve hot.