Cut asparagus into 1/2-inch lengths. In a 1 1/2- to 2-quart pan over high heat, bring 1 quart water to a boil. Add asparagus and cook until bright green and just tender when pierced, about 2 minutes. Drain, rinse with cold water, and drain again.
Meanwhile, in a large bowl, mix egg yolks, 1 1/4 cups cheese, milk, flour, cayenne, and salt until well blended. Stir in asparagus.
In another large bowl, with a mixer on medium speed, beat egg whites until stiff peaks form. Spoon about a quarter of the whites into yolk mixture; stir to combine. Scrape yolk mixture into bowl with remaining whites and fold together gently just to combine; do not overmix.
In an 11- to 12-inch nonstick, ovenproof frying pan over medium-high heat, melt butter; swirl pan to coat. Pour in egg mixture and smooth top with a heatproof, flexible spatula. Cook until edges appear set and dry and bottom is browned (loosen edges with spatula to check), 7 to 8 minutes.
Transfer pan to oven and broil 4 to 6 inches from heat until omelet is puffed, set, and lightly browned, 3 to 5 minutes. Sprinkle evenly with remaining 1/4 cup cheese; broil until cheese is melted and bubbling, 1 minute longer. Serve at once from pan, spooning onto plates.