Dice the eggplant and zucchini into a small dice.
Add butter to a medium-sized nonstick saute pan over medium-high heat. Heat one tablespoon of butter until it is melted and starts to sizzle. Add in the vegetables and allow to cook without stirring them until the veggies start to brown, about 3-4 minutes.
Season the vegetables with ¼ teaspoon of kosher salt. Stir the veggies and allow to cook until vegetables are translucent and softened about 1-2 minutes. Remove vegetables to a plate, dividing into portions.
Add another tablespoon of butter to the already hot pan and then crack in the eggs. Season the eggs with 1/8 teaspoon of salt and ¼ teaspoon of pepper.
Cover the eggs with a lid, turn the heat down to low, and cook for 1-2 minutes until the whites are cooked and there is a thin white layer over the yolks. The yolks should still be runny on the inside.
Top each pile of cooked vegetables with an egg and serve hot!