Pair with toasted crusty bread, a fresh fruit tart, and American fries.
Cook asparagus in medium skillet of boiling water until just tender, about 3 minutes. Drain; return asparagus to skillet. Mix cheese and basil in small bowl. Whisk eggs in large bowl.
Melt 1½ tablespoons butter in heavy large skillet over medium heat. Add eggs; stir until eggs are almost set, about 1 minute. Add cheese mixture and stir until cheese melts and eggs are softly set, about 2 minutes.
Meanwhile, add 1tablespoon butter to asparagus in skillet and stir over medium heat. Divide asparagus among 4 plates; spoon eggs over and serve.