For the garlicky croutons: Preheat the oven to 250 degrees F.
Spread out the bread cubes in a single layer on a baking sheet. Drizzle with the garlic oil and sprinkle with salt and pepper. Bake, tossing the croutons once halfway through, until crisp, about 25 minutes. Set aside and cool completely.
For the Caesar dressing: Put the garlic oil in a blender. Add the anchovies, mustard, vinegar, Worcestershire sauce, lemon juice, salt and pepper. Blend until smooth. Add the Parmesan and give it a quick blitz. (You want to retain some of the texture of the cheese.)
For the salad: Peel and pit the avocados, then slice lengthwise into spears and sprinkle with some lemon juice to prevent browning. Set aside.
Arrange the romaine in a line on a platter with the inside of each quarter-head of lettuce facing upward. Drizzle the dressing over the lettuce, allowing it to seep into the layers of leaves. Arrange the shrimp on top, along with the croutons. Scatter the cheese shavings and parsley leaves over the top. Squeeze the lemon half over the whole platter and serve.