Take the Chimichurri Sauce out of the fridge. Place it close to the stove top where you’ll cook so the heat will help warm it up and the oil will melt.
In a bowl, add the eggs, milk, and a sprinkle of salt and pepper and mix together.
Turn the broiler on high and move the oven rack so your pan will be about 4-5 inches from the broiler. Shred the chicken with two forks and set aside. Shred your cheese.
In a large cast iron pan (or oven safe skillet) heat the oil over medium heat. Add the green chilis and cook for two minutes, stirring often. Spread the black beans and chicken evenly over the entire pan. Add the egg mixture to the pan. As the egg mixture sets, run the spatula around the edge of the pan, lifting the egg mixture from the edges so the uncooked portion continues to flow underneath. Continue cooking and lifting the edges until egg mixture is almost set (the surface should still be very moist).
Remove from heat and sprinkle with cheese. Place the pan under the broiler and broil 1-2 minutes (or until the top has just set and cheese is melted).
Remove from the oven, cut slices, and drizzle with the Chimichurri Sauce and Green onions. Serve with avocado, arugula, and bread.

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