Chop together anchovy fillets, garlic, and pinch of salt. Use the side of a knife blade to mash into a paste, then scrape into a medium bowl. Whisk in egg yolks, 2 Tbsp. lemon juice, and mustard. Adding drop by drop to start, gradually whisk in olive oil, then vegetable oil; whisk until dressing is thick and glossy. Whisk in Parmesan. Season with salt, pepper, and more lemon juice, if desired.
DO AHEAD: Can be made 1 day ahead.
Make your own. Tearing, not cutting the bread ensures nooks and crannies that catch the dressing and add texture. Preheat oven to 375°. Toss bread with olive oil on a baking sheet; season with salt and pepper. Bake, tossing occasionally, until golden, 10–15 minutes.
Use whole leaves or tear by hand; they provide the ideal mix of crispness, surface area, and structure.
Caesars crowned with a mound of grated Parmesan may look impressive, but all that clumpy cheese mutes the dressing. Instead, use a vegetable peeler to thinly shave a modest amount on top for little salty bursts.
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