Preheat the oven to 400 degrees F. Soak the anchovies in a bowl of water, 5 minutes. Drain and pat dry, then finely chop and mash to form a paste; set aside.
Make the croutons: Combine the butter with the garlic in a small saucepan over medium-low heat; cook, swirling the pan occasionally, 5 minutes. Remove from the heat and pour into a medium bowl. Add the bread cubes and parsley and toss; season with salt and pepper. Arrange the bread on a baking sheet in a single layer and bake until golden brown and crisp, about 10 minutes; let cool.
Meanwhile, make the dressing: Bring a small saucepan of water to a boil. Gently lower the egg into the water, then immediately remove the pan from the heat and let stand 5 minutes. Drain and rinse the egg under cold water; crack into a large bowl. Whisk in the anchovy paste, the vinegar, Worcestershire sauce, lemon juice and mustard powder until smooth. Whisk in the olive oil in a slow, steady stream until emulsified; season with salt and pepper.
Add the lettuce, croutons and about one-third of the Parmesan to the bowl with the dressing and gently toss. Divide among plates and top with the remaining Parmesan.