How to Make It
Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Add the sausage and cook, breaking it up with a spoon, until browned and cooked through, 6 to 8 minutes; transfer to a plate.
Add the onion, sweet potatoes, thyme, and ¼ teaspoon each salt and pepper to the skillet and cook, stirring occasionally, until the onion is softened, 3 to 5 minutes (if the pan becomes too dark, add 2 tablespoons water and continue cooking). Add ½ cup of the broth, cover, and cook until the potatoes are tender, 5 to 7 minutes more.
Uncover the skillet and add the remaining ½ cup of broth. Add as much kale to the skillet as will fit and cook, tossing frequently and adding more kale when there is room, until the liquid is almost evaporated and the kale is tender, 4 to 5 minutes. Return the sausage to the skillet and toss to combine.
Meanwhile, heat the remaining tablespoon of oil in a large nonstick skillet over medium heat. Crack the eggs into the skillet and cook, covered, to the desired doneness, 2 to 4 minutes for slightly runny sunny-side-up eggs. Serve with the hash.