Position oven rack to center position. Preheat oven to 450°F and line a large, rimmed baking sheet with parchment paper and set aside.
Crack and separate eggs, placing the whites in a large mixing bowl and reserving the yolks in another bowl.* Season egg whites with salt and pepper, to taste.
Using a hand mixer or immersion blender, beat the egg whites until stiff peaks form. Scrape the sides of the bowl with a spatula while blending to incorporate all of the egg white.
Gently fold in 2/3 of the Asiago cheese and chives into the bowl, being careful not to deflate the egg whites in the process. Spoon the whipped egg white mixture onto the prepared baking sheet into four piles, leaving plenty of room between each mound. With the back of a spoon, create an indent in the center of each mound and place baking sheet on center rack in preheated oven and bake for 3 minutes.
After 3 minutes, remove from oven and transfer reserved yolks into the indent of each mount. Sprinkle each mound with half of the remaining cheese and fresh chives and return to oven and bake another 3 minutes or until yolks are set and egg whites are golden brown.
Remove egg clouds from oven and sprinkle with remaining cheese and chives and serve immediately.
*For best results, reserve the egg yolks in small, individual dishes rather than in one larger bowl.