Beat eggs together with milk, salt, and pepper in a bowl.
Heat oil in a skillet over medium heat. Add mushrooms; cook and stir until softened, about 5 minutes. Mix in green onions; cook until slightly softened, about 1 minute. Pour in the beaten eggs. Cook and stir until the edges harden, 2 to 3 minutes. Flip and sprinkle Cheddar cheese on top. Cook until egg is set and cheese is melted, 1 to 2 minutes. Fold omelet in half; top with more Cheddar cheese.
Olive oil can be substituted for the coconut oil.
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