Whisk eggs and egg whites in a bowl with some salt and pepper.

Add about 1/3 of the oil in a skillet over medium heat. Add the broccoli and cook for about 4 minutes until broccoli is tender but still firm. Transfer broccoli to a plate and set aside.

Wipe out skillet. Add remaining oil to the skillet and place over low heat. Pour in the eggs and gently stir to scramble until curds start to form, about 1-2 minutes.

Add in broccoli and cheddar. Continue scrambling until eggs are no longer runny. Serve.