Make an anchovy-garlic paste. Drain 1 (2-ounce) can oil-packed anchovy fillets. Mince the anchovies and 2 garlic cloves together until the mixture is mostly smooth and the garlic is almost paste-like, about 3 minutes.

Whisk the egg yolks. Place 3 large egg yolks in a medium bowl and whisk until smooth.

Add the mustard. Add 1 teaspoon Dijon mustard and whisk until combined.

Add the anchovy-garlic paste. Add the anchovy-garlic mixture and whisk until combined.

Whisk in the lemon juice. Juice 1 medium lemon until you have 2 tablespoons juice. While whisking, pour the lemon juice into the bowl and whisk until smooth.

Whisk in the olive oil. While whisking, slowly pour in 2 tablespoons olive oil to create a thick emulsion. Once all of the olive oil is added, whisk for another minute to thicken.

Finish with vegetable oil. While whisking continuously, slowly pour in ½ cup vegetable oil. Once all of the vegetable oil is added, whisk for another minute to thicken.

Season and serve. Add 2 tablespoons finely grated Parmesan cheese and whisk to combine. Taste and season with freshly ground black pepper as needed.