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1. Snap off and discard woody bases from asparagus. In a large bowl toss asparagus with olive oil, garlic, and salt and pepper to taste. Rub oil onto asparagus spears to coat evenly.
2. Grill asparagus, covered, directly over medium-hot coals for 5 minutes, turning once.
3. Meanwhile, heat a 12-inch skillet over medium heat. Add butter and wait until it foams. Add eggs to hot butter; sprinkle with salt and pepper. Reduce heat to low. Cook eggs for 3 to 4 minutes or until whites are completely set and yolks begin to thicken. Divide asparagus among four plates. Place two eggs on each mound of asparagus. Sprinkle with Parmesan cheese.