Place whole egg and egg yolk in a bowl of food processor and pulse to combine. Add anchovies that have been drained, and capers to the egg. Blend until pale/light yellow and frothy.
Add garlic and mustard. Blend, again.
Add lemon juice, salt, and pepper. Blend again.
With food processor running, pour in 1 1/2 cups of the oil in a slow drizzle/stream. Add parmesan. Continue blending. Add the rest of the oil in a slow drizzle.
Check consistency. If the dressing is too thick (like mayonnaise), add water, 1 tablespoon at a time, blending with each addition, until you reach the desired consistency. Personally, I like it thick like mayo, but make it to your desired consistency.
I like to add shredded (not grated) parmesan on my salad after I've mixed the lettuce with the dressing. Crumbled bacon is also good mixed in the salad.