Heat a medium size cast iron skillet or other skillet of your choice over medium heat. Add the red onion, a small splash of water and a pinch of the sea salt. Cook for a few minutes, until they start to soften. Add the carrots, and another pinch of sea salt and splash of water. Cook for another few minutes, or until all vegetables are tender.
Crack all of the eggs into a large bowl. Beat the eggs. Add the Dijon mustard, sea salt and black pepper. Stir again.
Preheat the oven to 350 degrees F.
Prepare your muffin tin by generously greasing each muffin cup with some coconut oil or avocado oil. Add a spoonful of the cooked red onion-carrot mixture to each cup, evenly distributing the veggies among the 12 muffin cups.
Pour some of the egg mixture into each muffin cup, dividing equally. Bake for 25 minutes or until the tops are set and no longer wet. Cool for 15 minutes. Use a knife around the edges to release the egg muffin cups from the muffin tin.