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In a 10 inch to 12 inch saute pan, heat 4 tablespoons butter until very dark brown. Add asparagus and toss until warm. Remove asparagus and divide among 4 plates. Towel off saute pan and add remaining butter. Cook until foam subsides. Crack eggs in and cook sunnyside up, about 2 minutes. Place one egg over each plate, sprinkle with cheese and serve.