In a large skillet, heat 3 tablespoons EVOO, 3 turns of the pan, over medium heat. Add the garlic and anchovy paste and cook until the garlic is softened, about 2 minutes. Add the escarole, lemon peel and nutmeg; season with pepper and cook for 2 minutes more.
In a large nonstick skillet, heat the remaining 2 tablespoons EVOO, 2 turns of the pan, over medium-high heat. Add the prosciutto and fry until crisp, 1 to 2 minutes. Transfer to a paper-towel-lined plate. Crack the eggs into the skillet and fry over medium-low heat to desired doneness; season with salt and pepper.
Stir the lemon juice into the escarole. Pile onto 4 plates and top each with 2 eggs and crispy prosciutto. Serve with the bread.