Dressing can be made 1 day ahead. Store in an airtight container and chill.
Chop together anchovy fillets, garlic clove, and a pinch of kosher salt. Use the side of a knife blade to mash into a smooth paste, then scrape into a medium bowl.
The paste should be very fine — keep working it until no chunky pieces remain.
Whisk in egg yolks, lemon juice, and Dijon mustard until combined.
Adding drop by drop to start, gradually whisk in the olive oil, then the vegetable oil. Continue whisking until dressing is thick and glossy.
Starting slow is key — adding oil too fast will break the emulsion.
Whisk in finely grated Parmesan. Season with salt, freshly ground pepper, and more lemon juice to taste.