Cook the egg in a pan of simmering water for 10 minutes. Cool under cold running water and peel.
Meanwhile, combine the anchovies, garlic, lemon juice and zest, and egg yolks in the bowl of a food processor and process to blend. Add the water and blend again. Then, with the motor running, gradually pour in the oil through the feed tube and process to emulsify. Scrape out into a salad bowl. Stir in the cheese and season with salt and pepper. Add the greens and toss to coat. Taste, for seasoning, and grate the egg over the top.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.