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Add chopped bacon to large, cast iron skillet, or non-stick skillet over medium-high heat.
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Cook bacon until crispy and remove from pan.
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Add onion, garlic, and rosemary to pan, cook until they begin to brown (stir occasionally).
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Add shredded potatoes, salt, and pepper. Spread into an even layer and cook undisturbed for 4 minutes.
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Reduce heat to medium and mix in Brussel Sprouts, stir occasionally and cook until the mixture begins to turn golden brown.
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Meanwhile, bring 2 inches of water to a boil in a medium saucepan.
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Reduce to a gentle simmer and stir the water (the circular motion helps to bring the egg and yolk together).
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Break an egg into a small bowl, submerge the bowl's lip in the water and slide the egg in.
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Transfer the egg to a paper towel to drain excess water.
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Repeat with the three remaining eggs.
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Serve the eggs over the hash.