Combine potatoes and 2 tablespoons water in a microwave-safe dish. Cover and cook on high for 4 minutes. Add Brussels sprouts, stir, and cook on high for another 2-3 minutes. The Brussels sprouts may not be completely thawed. This is fine–they will finish cooking in the skillet.
Meanwhile, heat oil in a large skillet over medium heat. Add onion and cook until softened, about 5 minutes. Stir in garlic or garlic powder and optional pepper flakes. Add Brussels sprouts and potatoes. Stir once, then let cook without stirring until the potatoes are browned on one side, about 5 minutes. Stir and allow to brown, undisturbed, 5 minutes more. Add salt and pepper to taste, remove from the heat, and set aside.
Melt butter in a nonstick pan over medium heat. Add eggs and cook until set, about 2 minutes, then flip and cook to desired degree of doneness, 1-2 minutes more.
Serve hash topped with eggs.
