Cut onion into thin half moons. Wash asparagus well, trim the ends and cut into chunks of about 1/2 to 3/4 an inch.
Heat oil in a large skillet and add onions. Cook until brown. Add asparagus and cook until tender crisp (about five minutes). Season with salt and pepper and set aside.
Grease cups/jars. Using tongs, portion out the sautéed vegetables. The cups/jars should be mostly filled with vegetables.
Beat eggs and evenly divide between the baking cups/jars. Top with approximately 1 tablespoon of cheese per serving. Using a fork or spoon, carefully stir the contents of each cup/jar so that the cheese is not only resting on top.
Place the cups/jars on a baking sheet and bake for 20-25 minutes. When tops are browned, they are done.
Remove from oven and let cool. If used jars, put the lids on once they are cool enough to touch (the tops will be puffy and high just after baking, but like a soufflé, they will fall). Store egg cups in the refrigerator until ready to eat. They will keep up to five days.