Scoop some avocado where the seed was to make a larger well for the egg. Just eat what you scooped out.
In a heavy bottom skillet or pot with a fitted lid, heat oil. Add avocado halves taking care not to let them break as well as tomatoes. Allow to cook in the hot oil 3-5 minutes.
Crack one egg into each half and add 3-4 tablespoons of water to the bottom of the pan. Be careful as the water might splatter in the hot oil. Cover tightly with lid to poach the eggs with the steam.
After 5-10 minutes, check doneness of eggs by jiggling them. You want the yolks to remain runny while the whites get fully cooked.
Use a spatula to transfer the avocado eggs to a plate. Add basil to the tomatoes in the pan and stir to combine, the top eggs with tomato mixture. Season with salt and pepper, as desired.