Preheat the oven to 375°F. Grease muffin tins (enough for 14-16 large muffins), and set aside.
Steam or cook the broccoli florets until tender. Remove from heat and set aside. In a large bowl crack the eggs and add the egg whites and whisk them together. Set aside.
In a large skillet, heat 2 tablespoons olive oil over medium high heat. Add the onion and bell pepper and sauté for 6-7 minutes, until the onion is soft.
Add the cooked vegetables and broccoli to the eggs along with 1 cup of the cheddar cheese. Mix to combine. Add the nutmeg, salt and pepper and gently stir to combine everything together.
Pour the egg mixture into the greased muffin tins (about ¾ of the way full) and sprinkle each muffin with a small amount of the ½ cup of cheddar cheese.
Put the egg muffins in the oven and bake for 20-25 minutes until the egg muffins are set in the middle and begin to turn golden brown on the edges.
Remove from the oven and let cool for a ½ hour before removing from the muffin tin.
Refrigerate the egg muffins in an airtight container for up to one week.