Remove the stem and outer leaves of the lettuce. Carefully separate the inner leaves, rinse and dry them with a clean tea towel. Rest them in the fridge until you’re ready to serve. (If you have time and/or want really crispy leaves, rest them for a few hours or overnight)
Preheat the oven to 170 degrees.
Put the egg yolks, lemon juice, vinegar, mustard, garlic & anchovy in a food processor and pulse for a few seconds. Add a pinch of salt, then slowly drizzle (alternating) the vegetable and olive oil, while the food processor is very slowly blending the mixture into mayonnaise.
When you get a smooth, silky mayonnaise, add 1-2 tablespoons of water to thin it. Taste it, and adjust the seasoning and vinegar to your liking. Rest in the fridge.
Cut the bread into small cubes or break the into bite-sized pieces with your hands (if you want a rustic version).
Rub the bottom of a baking tray with garlic, then leave the garlic in the pan. Scatter the croutons evenly in the tray, season with salt and pepper, sprinkle with thyme leave (if used), then drizzle or spray them with some oil olive oil. Bake the croutons in the oven for about 10 minutes until crisp and golden. Let them cool.
Arrange the lettuce leaves in a large serving dish, scatter the croutons all over. Using a potato peeler, shave some Parmesan pieces onto the top. Finally, drizzle with the mayo dressing, season with salt&pepper, then serve.