In a large nonstick skillet over medium heat, fry bacon until crispy.
Turn off heat and transfer bacon to a paper towel-lined plate. Keep most of bacon fat in skillet, removing any black pieces from the bacon.
Turn heat back to medium and add onion, bell pepper, and cauliflower to the skillet. Cook, stirring occasionally, until the vegetables begin to soften and turn golden. Season with salt, pepper, and paprika.
Add 2 tablespoons of water and cover the skillet. Cook until the cauliflower is tender and the water has evaporated, about 5 minutes. (If all the water evaporates before the cauliflower is tender, add more water to the skillet and cover for a couple minutes more.)
Take off the lid, then stir in the garlic and chives and cook until the garlic is fragrant, about 30 seconds. Using a wooden spoon, make four holes in the hash to reveal bottom of skillet. Crack an egg into each hole and season each egg with salt and pepper. Sprinkle cheese and cooked bacon bits over the entire skillet. Replace lid and cook until eggs are cooked to your liking, about 5 minutes for a just runny egg. Serve warm.
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