Chop together 3-6 anchovy fillets packed in oil, drained, 1 small garlic clove, and a pinch of kosher salt* (*after, to taste—if desired). Use the side of a knife blade to mash into an anchovy paste, then scrape into a medium bowl.
Whisk in 2 large egg yolks, 2 Tbsp. fresh lemon juice, ¾ tsp. Dijon mustard, and 1 tsp Worcestershire sauce.
Adding drop by drop to start, gradually whisk in 2 Tbsp. extra-virgin olive oil, then ½ cup vegetable oil; whisk until dressing is thick and glossy.
Whisk in 3 Tbsp. finely grated Parmesan. Season with salt, freshly ground pepper, and more lemon juice, if desired.
Do ahead: Dressing can be made 1 day ahead. Store in an airtight container and chill.
Preheat oven to 375°.
Toss 3 cups torn 1" pieces country bread, with crusts, with 3 Tbsp. extra-virgin olive oil on a baking sheet; season with kosher salt and freshly ground pepper.
Bake, tossing occasionally, until golden, 10–15 minutes. Do ahead: Croutons can be made 2 days ahead. Store in an airtight container at room temperature.
Place 3 romaine hearts, leaves separated, in a large bowl.
Use a vegetable peeler to thinly shave a modest amount of Parmesan on a plate.
Use your hands to gently toss together the lettuce, croutons, and dressing. Top with the shaved parm.
