Place bacon in a large skillet (frying pan) and cook over medium-high heat, turning occasionally, until crispy, 5 to 8 minutes. Drain on paper towels; reserve most of the bacon fat in skillet.
Reduce heat to medium. Cook and stir onions until starting to soften, about 5 minutes. Add potatoes; toss to coat evenly in bacon fat. Pour in water and cover skillet with a large lid. Cook, checking occasionally, until potatoes are tender, about 20 minutes. Add green onions, garlic, and paprika. Season with salt and pepper.
Make 4 wells in the potato mixture using a wooden spoon, revealing the bottom of the skillet. Crack an egg into each well; season with salt and pepper. Sprinkle cooked bacon and Cheddar cheese over the entire skillet. Replace lid; cook until egg whites are set and yolks are still runny, about 5 minutes.
If potatoes start to brown too quickly, add more water.