Hand-tear the bread into small pieces and put them into a bowl. Add the olive oil, Parmesan, and thyme leaves; season with salt and pepper. Mix this well to combine all the flavors. Spread the cubes out onto a baking sheet and bake until they are golden brown, about 15 minutes. Let them cool before tossing them onto the salad.
Make the dressing: Put the anchovies, garlic, lemon juice, and water into a blender and process for 30 seconds until the mixture is smooth. With the blender running, pour the olive oil in slowly for the dressing to emulsify. Stir in the Parmesan and a couple of grinds of black pepper. Season, to taste, with salt and set aside. (Refrigerate the dressing if you will not be using it right away.)
Assemble the salad: Tear the lettuce into a large bowl. Add enough dressing to coat the salad to your liking. Add the croutons and some extra Parmesan and toss the salad well. Serve immediately.