Toss asparagus with a little olive oil; arrange in a single layer on a rimmed baking sheet and sprinkle with salt and pepper. Roast 8 to 10 minutes.
Meanwhile, in a small bowl, combine cheese and basil. In a large bowl, whisk eggs.
Melt the butter in a large skillet over medium heat. Add eggs and stir until they are almost set, about 1 minute. Add cheese mixture and stir until cheese melts and eggs are softly set, another 2 minutes.
To serve, place asparagus on individual plates and serve eggs spooned over asparagus. Toast and butter English muffins and serve with the eggs.
How to trim asparagus: Check for a tough inedible portion at the base. Simply break off the tough end of each spear by bending it gently until it snaps. It will break exactly where the tender part begins.