Preheat oven to 375°F (190°C).
Heat oil in a small pan over medium heat. Add chopped onion and saute for 3-4 minutes until starting to turn translucent. Add broccoli and garlic and saute another 2-3 minutes to soften, stirring to evenly cook.
In a medium to large bowl, beat eggs and milk together using an electric mixer or wire whisk. Add parsley, salt, pepper, hot sauce, and mustard and beat to mix well.
If using a metal mini-muffin tin, grease each muffin cup with cooking spray or rub with oil (be thorough!). This step is not needed if using silicone cookware. If using silicone cookware, make sure to place a cookie sheet underneath to avoid spillage and make it each to remove later on.
Split veggie mixture between each prepared mini-muffin cup (each should be approximately 1/3 full).
Sprinkle cheese on top of veggies.
Pour beaten egg mixture over veggies, leaving a small amount of space at the top to avoid overflow while baking.
Place in oven and bake for 12 - 14 minutes until set. Don't over bake or eggs will be rubbery!
Remove from oven and let cool for 5-10 minutes before tracing around each egg muffin with a butter knife too loosen and remove from pan.
Serve immediately or store in an airtight container in the fridge and reheat using the microwave. These egg muffins will last for 4-5 days.