In a small bowl, stir together the panko, Parmesan, vinegar, sesame seeds, thyme, Aleppo pepper, ¼ teaspoon salt and ½ teaspoon black pepper. In a 12-inch nonstick skillet over medium, melt 2 tablespoons (4 pieces) of butter. Add the panko mixture and cook, stirring, until light golden brown, about 3 minutes.
Working quickly, use the back of a spoon to create 4 evenly spaced clearings in the panko. Drop 1 piece of the remaining butter into each and let melt, then crack an egg into each. Sprinkle the eggs with salt and black pepper, then cover the pan and cook, occasionally rotating for even browning, until the whites are set but the yolks are runny, about 5 minutes.
Remove the pan from the heat. Using a thin, wide spatula, transfer the eggs with breadcrumbs to individual plates. Sprinkle on any breadcrumbs remaining in the skillet. Sprinkle with chives (if using).
