Mash the anchovies and grated garlic together into a smooth paste in a medium bowl.
Whisk in the egg yolks, Dijon mustard, lemon juice, Worcestershire sauce, salt, and pepper until smooth.
While whisking constantly, slowly drizzle in the neutral oil until the dressing emulsifies and begins to thicken.
Continue whisking while slowly adding the olive oil.
Stir in the Parmigiano-Reggiano until fully incorporated.
Taste and adjust with additional lemon juice, Parmesan, or salt as needed. If you'd like a thinner dressing, whisk in 1–2 tablespoons of cold water.
Refrigerate for 30 minutes before using to allow the flavors to meld.