Fill a pot with heavy bottom with cool water until it’s about 1 to 2 inches above the eggs. Close the vent lid and heat the water until it just starts to boil. Lower the heat, cover the pot fully, and let it boil for 30 seconds. Take it off the heat and let it sit for 12 minutes with the lid on. Put the eggs in a bowl of ice water and let them sit for 10 minutes. Then, run cool water over them to peel them. Cut eggs in half down the middle.

Scoop the yolks into a bowl with care. Leave the egg whites alone. Add garlic, lemon juice, mayonnaise, and 1 teaspoon of extra-virgin olive oil. Mix with a hand mixer or a whisk until the mixture is smooth. Add the fresh basil and season with salt, pepper, and more lemon juice, if you want.

Warm up the last ½ teaspoon of olive oil in a small pan over medium-low heat. Mix in the bread crumbs and stir until the crumbs are slightly toasted. Add a little salt and black pepper to a bowl after moving the food there. Allow to cool. Add the lemon juice and Parmigiano Reggiano and mix well.

Fill egg white halves with a mixture of garlic and basil yolks. Before serving, sprinkle breadcrumbs on top of each egg, and serve right away after putting them together.