Position a rack in the middle of the oven and preheat to 400 degrees. Set a kettle of water to boil.

Arrange the bread slices on a large, ungreased sheet pan and toast for about 10 minutes, or until crisp. Brush both sides of the bread with 1 ½ tablespoons of the oil and return to the oven for another 4 minutes, or until golden brown, flipping halfway through.

Meanwhile, on a cutting board, mince and mash the garlic and anchovies together into a mostly smooth paste; you can use the side of the knife to help press and mash them. Transfer the mixture to a small bowl and gradually add 1 tablespoon of the oil, using a fork to mix and mash until incorporated. Spread the mixture onto one side of each bread slice. (Applying it to the hot bread will help mellow the flavors, but you may need to wait 1 to 2 minutes, until the slices are cool enough to handle.)

Place the egg in a small bowl and pour over enough of the boiling water to cover it. Let rest to partially cook for 3 minutes. Drain, and crack the egg into the same bowl or a liquid measuring cup; the white should be opaque and just setting. Lightly beat the egg with a whisk or fork until mostly combined.

In a large bowl, combine the lettuce, parmesan, the remaining 1 ½ tablespoons of oil, the lime juice and Worcestershire sauce, and gently toss with the bread slices and egg until well incorporated. Season to taste with salt and pepper, and serve immediately.