Heat a medium-sized skillet over medium heat. Melt butter and then add sprouts, shallots, and salt. Heat until sprouts are tender and slightly browned, 6 to 8 minutes. Stir in spinach, garlic, and lemon zest; heat until spinach is lightly wilted.
Create two nests in the skillet for the eggs. Crack eggs into nests, cover skillet, reduce heat to low and cook until egg whites are set and yolks have reached desired consistency, about 10 to 12 minutes for runny yolks.
Season with black pepper and scatter on bread crumbs, goat cheese, and walnuts.