Add the anchovy paste, mayonnaise, lemon juice, and balsamic vinegar, and pulse to combine. Taste for salt, garlic, and lemon juice balance.
Starting with the olive oil, begin adding the oil through the lid at a slow stream, pulsing the processor on/off to avoid heating and breaking the emulsion. Stream the canola oil after all the olive oil has been added.
Add Parmesan cheese and black pepper, then pulse to combine. Adjust the seasonings if needed.
Method 2: Make by hand
Place salt and garlic (and anchovy fillets, if using) on a cutting board. Using the flat side of a chef’s knife, carefully smash ingredients together until mixture becomes a thick paste and is thoroughly blended. Transfer paste to a small bowl.
Add anchovy paste (if using), mayonnaise, lemon juice, and vinegar to the paste, and whisk until blended.
Add oil (first olive oil, then canola oil) in a slow, steady stream, whisking constantly until smooth.
Whisk in parmesan cheese, and season with black pepper to taste.
Transfer to a glass jar and serve. Dressing can be stored in the refrigerater for up to 1 week.
I use two 2-inch "lines" of anchovy paste, so I'm guessing it's about 1 teaspoon. Use more or less to suit your taste.If you don't have (or don't like) anchovies, you can substitute ½ teaspoon Worcestershire sauce (or to taste), or just leave it out altogether.You can use two large pasteurized egg yolks in place of the mayonnaise, if you wish. I just find mayonnaise to me easier (and safer).