Place the bacon in a cold skillet. Turn the heat to medium and cook until crispy, stirring occasionally. Remove from pan and drain on paper towels.
Add the butter to the pan, melt, and add the kale in handfuls, letting it wilt down then adding more. Season with a large pinch of salt and pepper. Cook, stirring often until fully wilted. Turn up the heat, add the vinegar and cook for 1 minute or until the vinegar has evaporated. Add the broth, stir, and cook for another minute.
Make 4 wells in the kale with the back of a spoon. Crack and egg into each well, sprinkle each egg with a little salt and pepper, turn the heat down to medium, cover the pan and cook until the eggs are cooked to your liking, about 4 minutes for a cooked white and runny yolk.