Squeeze the fresh lemon juice, finely grate the Parmesan, and grate the garlic into a smooth paste. Finely grating the garlic helps prevent large, sharp-tasting pieces from ending up in the dressing.
Add 3/4 cup (170 g) mayonnaise, 2 tablespoons (30 ml) fresh lemon juice, and 1 teaspoon (5 ml) Dijon mustard to a medium mixing bowl. Whisk until completely smooth, with no visible streaks of mustard or mayonnaise.
Stir in 2 tablespoons (10 g) finely grated Parmesan, 1 teaspoon (5 ml) Worcestershire sauce, the grated garlic clove, and 1 teaspoon (5 ml) anchovy paste if using. Whisk until the cheese and garlic are evenly distributed. The dressing should look creamy with tiny flecks of Parmesan.
Whisk in 1 tablespoon (15 ml) cold water: The dressing should be thick enough to coat the back of a spoon but loose enough to drizzle. If it still feels too thick, add more cold water, 1 teaspoon (5 ml) at a time, until it reaches the desired consistency.
Stir in 1/4 teaspoon (about 0.5 g) freshly ground black pepper and 1/8 teaspoon (about 0.75 g) fine salt. Taste and adjust the salt if necessary. The dressing should taste tangy and savory, with a noticeable but balanced garlic flavor.
Transfer the dressing to a clean airtight jar or container and refrigerate for at least 30 minutes. It may thicken slightly as it chills, and the garlic flavor will become more noticeable.
Stir the chilled dressing until smooth: If needed, loosen it with a little additional cold water. Toss with crisp romaine lettuce and your preferred Caesar salad ingredients, or serve it separately so everyone can add their desired amount.