How to Make It
Prepare the croutons: Preheat oven to 400°F. Heat olive oil and garlic cloves in a small saucepan over medium-high; cook until garlic begins to sizzle, 4 to 5 minutes. Remove and discard garlic.
Place cubed baguette in a medium bowl; drizzle with garlic-infused oil. Toss to coat. Sprinkle with Parmesan, salt, and pepper. Toss to coat. Arrange in a single layer on a rimmed baking sheet. Bake in preheated oven until toasted, 8 to 10 minutes. Set aside.
Prepare the salad: Heat vegetable oil in a heavy saucepan over medium-high to 375°F. Fry capers in 2 batches until golden, 30 seconds per batch. Using a slotted spoon, transfer to paper towels to drain.
Combine vinegar, lemon juice, water, mustard, salt, anchovy fillets, garlic, and egg yolk in a blender. Process until smooth. With blender running, slowly add olive oil until thickened. Transfer dressing to a bowl; stir in pepper and ¼ cup of the grated cheese.
Combine romaine hearts, croutons, and remaining ¼ cup grated cheese in a large bowl; drizzle with ¾ cup of the dressing, and toss gently to coat. Divide lettuce mixture among 4 plates; sprinkle with capers and shaved cheese. Serve with remaining dressing on the side.