Whisk anchovies, garlic, lemon juice, oil, 2 tbsp. water, the salt, and pepper in a very large bowl. Don't worry that the dressing doesn't emulsify; it will come together more when you add the cheese.
Add 3/4 cup parmesan and the greens and mix with your hands. Set salad on plates. Add more parmesan if you like.
Note: Nutritional analysis is per serving.