Preheat the oven to 400 degrees. Cut 3 cloves of garlic into 1/8-inch thick slices and reserve the 4th clove. In a small saucepan or skillet set over moderately low heat, combine the sliced garlic with 1/2-cup of the oil and cook for 10 minutes. Strain the oil into a bowl and reserve the cooked garlic.
In a small bowl, whisk together the vinegar and mustard, add 1 teaspoon of the anchovy, and mash with the whisk until smooth. Whisk in the garlic-flavored oil, the cooked garlic, and remaining anchovy. Season with salt and pepper.
Brush both sides of the bread with the remaining 1/4 cup oil and season with salt and pepper. Arrange the bread in a single layer on a baking sheet and bake until lightly toasted -- about 5 minutes. When cool enough to handle, rub one side of each slice with the remaining raw garlic clove.
Peel the eggs and separate the yolks from the whites. With the back of a spoon or a rubber spatula, push the yolks through a fine sieve into a bowl. Wash and dry the strainer and repeat with the whites.
Choose 6 large Romaine spears and slice them crosswise into 1/4-inch thick ribbons. In a large bowl, toss the remaining whole leaves with the dressing. Arrange the leaves on salad plates. Toss the lettuce ribbons in the residual dressing and set the ribbons on top of the spears. Sprinkle with the whites and yolks and arrange the toast on the plate. Top the toast with the Parmesan shavings.