Preheat oven to 350 degrees. Lay baguette slices on a large rimmed baking sheet. Brush bread on both sides with 2 tablespoons oil; season with salt and pepper. Bake until golden, 10 to 15 minutes, turning over halfway through; let croutons cool on baking sheet.
In a blender, combine lemon juice, Parmesan, mayonnaise, anchovies, garlic, 4 reserved hard-cooked egg yolks, and remaining 2 tablespoons oil. Season with salt and pepper; blend dressing until smooth.
In a large bowl, gently toss together lettuce, croutons, dressing, shaved Parmesan, and remaining whites and eggs.