Get things ready! Chop up the Brussels sprouts until they they are finely shredded. Fill a pot with a few inches of water, and place it over a hot burner.
In a large saucepan, melt the butter. Add the shredded Brussels sprouts and cook until they are soft but still al dente (7-10 minutes). Add the peas and cook until they are warm. Salt and pepper to taste, and distribute in 2 servings.
Add a splash of vinegar to the water, and crack an egg into a small cup. When the water is just on the verge of boiling (but not quite), take a spoon and make a whirlpool in the hot water. Drop the egg very gently into the center of the whirlpool and allow to cook for 3-4 minutes. When the egg is cooked, scoop it up with a slotted spoon and set it to drain on a paper towel. Poach the remaining eggs.
Serve each pile of hash with 2 eggs and sprinkle each plate generously with Pecorino Romano.