In a medium bowl, add the peeled and seeded avocado, 2 tablespoons lime juice (or to taste), 1 to 2 tablespoons finely minced chives, 3/4 teaspoon kosher salt (or to taste), and 3/4 teaspoon freshly ground black pepper (or to taste). Coarsely mash with a fork to your desired texture and set aside.
Use a 2.5- to 3-inch round cookie cutter or glass to cut a circle from the center of each slice of bread; reserve the cut-out rounds.
Heat a large nonstick skillet over medium-low heat. Add 2 tablespoons unsalted butter and melt, swirling the pan to coat the surface.
Place the two bread slices and the cut-out rounds in the skillet in a single layer. Cook until the bottoms are golden brown, about 1 to 2 minutes.
Flip the bread slices and the rounds. Crack one large egg into the hole of each flipped bread slice (one egg per slice). Season the eggs with salt and freshly ground black pepper to taste.
Cover the skillet and cook until the eggs reach your desired doneness, about 3 to 6 minutes (check at 3 minutes for soft yolks).
While the eggs cook, monitor the cut-out rounds—they will brown faster than the egged bread (about 1 to 2 minutes after flipping). Remove the rounds as soon as they are golden brown and place them on a plate.
When the eggs are cooked to your liking, remove the egg-in-a-holes from the skillet and place them on the plate with the rounds.
Evenly spread the avocado mixture over the egg-in-a-holes and the toasted rounds. Serve immediately while warm.