Place eggs in a bowl filled with hot water, not scalding. This will warm the eggs up slightly and allow for quicker cooking.
Place half of the butter into a medium non-stick skillet and turn heat to medium-low. Add mushrooms and cook for 2-3 minutes, or until nicely browned and softened. Remove mushrooms and set in a dish.
While the mushrooms are cooking, crack eggs into a bowl and add milk. Salt and pepper to taste. Using a fork, slightly stir the eggs but take care not to incorporate too much air into the eggs. Once the mushrooms are finished, place the remaining butter into the skillet and stir it around using a silicone spatula. Add the eggs and cook for about 2 minutes, just till they are almost done. Once they are mostly cooked through, add the cheese over the top of the eggs and the mushrooms. Cook for about 1 minute longer, or until eggs are finished, and then roll the omelet out of the pan and onto a plate. Sprinkle green onion on top and serve warm.