Trim off the root end and the tough upper greens from leeks. Cut the trimmed leeks in half lengthwise. Rinse in a bowl of cold water. Lift the leeks out of the water to strain. Heat a large sauté pan and add olive oil. Sauté the leeks with the cut-side down, season with salt and pepper and brown (approximately 2-3 minutes).
Add the thyme leaves, sliced lemon and water to pan. Cover the pan and reduce the heat to medium-low. Continue to braise for 10 minutes. Remove the leeks from the pan until they are cool to the touch.
Cut the leeks in half, place them in a casserole dish and top with chopped mozzarella. Bake the dish for 15 minutes, or until the cheese is melted and heated through. You can set the oven to broil for the last 2-3 minutes to brown the cheese.
While the leeks are in the oven, mix all mustard crumb ingredients together. Remove the casserole from the oven and set aside.
Fry your eggs in remaining oil, sunny-side up. Evenly distribute the eggs on top of the casserole and top with the mustard crumb situation. Burst the egg yolks with a teaspoon, drizzle the yokes over the dish and serve.