In a medium bowl, combine cheese, pimientos, and mayo. Stir well to combine.
In a separate bowl, whisk together the eggs and milk. If making more than one omelet, only do one at a time. Season with salt and pepper.
Lightly coat a nonstick pan with butter and heat over medium-high heat.
Pour egg mixture into pan and stir with a rubber spatula for 5 seconds. Begin pushing the cooked portions to center of pan and tilt pan to let uncooked portions run to sides. When no more uncooked portions run to the sides, let omelet cook undisturbed until top begins to set.
Add 1/4 to 1/3 cup pimiento cheese mixture along one side of the omelet. Sprinkle green onion on top of pimiento cheese. Fold other side over. Turn heat to low and let sit for about 1 minutes to melt cheese.