Cut the avocado in half, remove the pit and scoop out enough of the flesh to accommodate an entire egg.
Remove a small portion of the skin on the back so the avocado sits straight when you set it on the cutting board.
Crack the eggs and divide them between 3 containers. Place the yolks in individual shot glasses or small teacups and place both whites together in a common small mixing bowl. Add salt and pepper to taste to the whites and mix well.
Heat coconut oil in a skillet with fitting lid set over medium high heat. Add the avocado halves, flesh side down, and sear them for about 30 seconds, or until slightly golden.
Flip the avocados around and fill the cavities almost to the top with the egg whites. Turn the heat down, put the lid on and cook for about 15-20 minutes or until the egg whites are almost set.
Carefully slide the yolks over the whites and continue cooking for 3-5 minutes or until yolks have reached the desired level of doneness.
Transfer to a serving plate and garnish with walnuts, thyme and balsamic pearls.