Heat a grill to medium-high.
In a large skillet, heat the EVOO, two turns of the pan, over medium. Add the anchovies; stir until they melt into the oil, 2 to 3 minutes. Add the leek and garlic and stir until tender, about 4 minutes. Add the greens and stir until wilted, about 1 minute. Season with nutmeg and salt and pepper. In a small bowl, mix the ricotta, thyme and lemon zest.
Brush the flatbreads with EVOO. Working in batches if necessary, grill until the bottom of the bread is crisp, about 4 minutes. Turn the bread over and top with dollops of the ricotta mixture, greens mixture, mozzarella and Parmigiano-Reggiano. Cover and grill until the cheeses melt, about 5 minutes.
Wipe out the skillet. Add the butter and melt over medium heat. Crack the eggs into the skillet, season with salt and pepper, and cook until the whites are almost set, 2 to 3 minutes. Cover and cook until the whites are just set but the yolks are still runny, about 1 minute more. Divide the eggs among the pizzas. Sprinkle with coarsely ground black pepper.