Move your oven rack to the center position and preheat the oven to 450°F. Prepare a large, rimmed baking sheet by lining it with parchment paper. Set aside.
Crack and separate eggs. Place the egg whites in a large mixing bowl and reserve the yolks separately. (For best results, do not place egg yolks in one larger bowl; reserve them individually in small dishes.) To taste, season egg whites with salt and pepper.
Beat the egg whites until stiff peaks form using an immersion blender or hand mixer. Make sure to scrape the sides of the bowl with a spatula during this process to incorporate all egg whites.
Gently fold in ⅔ of each the chives and the Asiago cheese. Be careful not to deflate the beaten egg whites during this process.
Onto your baking sheet, spoon the egg white mixture into four equal-sized mounds, leaving plenty of room in between each one. Indent the center of each mound with the back of a spoon.
Place baking sheet into pre-heated oven on the center rack. Bake for 3 minutes.
Remove baking sheet from oven after 3 minutes and transfer one reserved egg yolk into the indent of each mound. Sprinkle half the remaining chives and Asiago cheese onto each mound.
Return baking sheet to oven and bake for another 3 minutes until the whites are a golden brown and the yolks are set.
Remove baking sheet from oven and sprinkle remaining chives and Asiago cheese on the egg clouds. Serve immediately.