Whisk the eggs in a large bowl. Whisk in the Calabrian chili oil and half of the chopped chives and season with salt and pepper.
Melt 2 tablespoons of the butter in a wide nonstick pan over medium-low heat. Add the eggs and cook, slowly agitating them with a rubber spatula to form large curds, 4 to 5 minutes. Turn the heat off 30 seconds before the eggs reach your desired doneness.
Halve the Parker House rolls and place them on a sheet tray. Spread the remaining 2 tablespoons of butter on each slice and toast until golden brown, 3 to 5 minutes.
Divide the eggs between the bottom halves of the rolls. Top with the grated Asiago cheese and broil until melted, 1 to 2 minutes. Top with the remaining half bunch of finely chopped fresh chives. Enjoy.