In a large cast iron or other oven-safe skillet over medium heat, heat the olive oil and butter. Add the shallots, and sauté for 2 to 3 minutes or until fragrant.
Add the Brussels sprouts and pears and cook until tender, stirring occasionally, about 10 to 15 minutes. Season with the salt and pepper throughout the recipe, and stir in the rosemary towards the end.
Preheat the broiler to high. Using the back of a spoon, make a couple wells for the eggs in the hash. Crack the eggs into them, then transfer the skillet to the center rack of the oven. Cook for 1 to 3 minutes or until the eggs are cooked to your liking (keep a close eye on them). Serve warm!
Check out the tutorial on how to shred Brussels sprouts if needed. You can also find pre-shredded Brussels sprouts in the produce aisle at most grocery stores, if you want to save time.
If you don't have an oven-safe skillet, cook the hash in a large skillet that comes with a lid. When you crack the eggs into the wells, cover the skillet and cook for a few more minutes on the stove until the eggs are cooked to your liking.
Possible additions include chopped bacon and cheese.