Bring 4 quarts water to boil and set up ice bath. Trim asparagus of hard ends and drop into water. Cook 70 seconds. Remove with tongs and refresh ice bath. Drain and set aside.
In a 10 to 12-inch sauté pan, heat 4 tablespoons butter until very dark brown. Add asparagus and toss until warm. Remove asparagus and divide among 4 plates.
Towel off sauté pan and add remaining butter. Cook until foam subsides. Crack eggs in and cook sunny side up about 2 minutes.
Place one egg over each plate, sprinkle with cheese and serve
