To hard boil the eggs, you can use any technique you like. This is ours: Fill a bowl with very hot tap water. Place the eggs in the bowl for 10 minutes to temper them. Meanwhile, bring a pot of water to a low boil. After the eggs have tempered, gently place them into the boiling water for 10 minutes. Remove the eggs to a bowl of cold water (not ice water) until they are cool enough to handle and peel. Peel the eggs carefully to keep the white intact.

Once peeled, halve the eggs lengthwise very carefully. Remove the yolks to a bowl with the mayonnaise, avocado, basil, lime zest, salt, and pepper. Mix well to eliminate any lumps and set aside.

Heat a skillet over medium heat and fry the prosciutto slice for about 20 seconds to crisp it up, render some of the white fat, and intensify its flavor. Place the prosciutto on a paper towel to cool for a few minutes and then chop the prosciutto into small pieces and add to the egg yolk mixture.

Ensure you have mashed the egg yolks and avocado thoroughly along with the rest of the ingredients. You can use a fork for this. Once the mixture is thoroughly mashed and combined, place the mixture into a piping bag or a zip-top bag that you can cut the corner off.

Pipe the mixture into the halved egg whites and garnish with chopped basil or a very small basil leaf and/or a reserved piece of chopped prosciutto. Serve immediately or refrigerate for service later that same day.