Position an oven rack in the middle of the oven and preheat to 300 degrees F.
Stir the butter, grated garlic and Italian seasoning in a small bowl until combined. Use a rolling pin to flatten out each slice of bread. Trim the edges to remove the crusts and make each slice into a 3-inch square. Brush both sides of each slice with the butter mixture. Press one slice into each cup of a 12-cup muffin pan, so that the bread is completely covering the bottom and the corners are coming up the sides of the cup.
Bake until golden brown, 20 to 30 minutes. Let cool in the muffin pan 15 minutes, then transfer the crouton cups to a rack to cool completely; they will crisp up as they cool.
Using a vegetable peeler, shave 12 small Parmesan curls onto a small plate. Finely grate the remaining Parmesan (you will need 3 tablespoons) and set aside.
Mound the anchovies, chopped garlic and 1 teaspoon salt on a cutting board. Using the side of a chef’s knife, mash and chop the anchovies and garlic until well combined, then continue to work the mixture, holding the knife blade at an angle, until a smooth paste forms.
Whisk the egg yolk, lemon juice and mustard together in a medium bowl. Adding drop by drop to start and whisking constantly, drizzle a few drops of oil into the yolk mixture. Continue, going slowly, until all the oil has been used and the mixture looks thick and creamy. Add the anchovy mixture and the grated Parmesan and whisk until smooth. Taste and adjust the seasoning with salt if needed.
Toss the lettuce, chicken and half of the dressing in a large bowl, then taste and add additional dressing if desired. Mound some salad inside each crouton cup, then top each with a Parmesan curl and serve.