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Heat oil in a large nonstick skillet over medium-high heat. Add onion and rosemary; cook, stirring, until beginning to brown, about 2 minutes. Stir in hash browns, salt and pepper. Spread into an even layer. Cook, undisturbed, for 4 minutes.
Reduce heat to medium. Stir in Brussels sprouts, spread back into an even layer and cook, stirring every 2 to 3 minutes and returning to an even layer, until golden brown, 12 to 14 minutes total. Remove from heat and stir in cheese.
Meanwhile, bring 2 inches of water to a boil in a Dutch oven. Reduce to a gentle simmer. Break an egg into a small bowl, submerge the bowl’s lip in the water and slide the egg in. Repeat with the remaining eggs. Cook for 4 minutes for soft set, 5 minutes for medium set and 8 minutes for hard set. Transfer the eggs to a clean dish towel to drain. Serve the eggs over the hash.