BEAT eggs and milk in a small mixing bowl; season lightly with salt and pepper. Set aside.
MELT ½ teaspoon butter in a small frying pan over low-medium heat. Add half the amount of broccolini pieces and cook. Carefully pour half the egg mixture over cooked broccolini. Stir with heat resistant rubber spatula until just cooked; approximately 30 seconds.
TURN off the heat and, using rubber spatula, smooth over the top of the omelet so that it evenly covers the inside of the small frying pan. Sprinkle 1 tablespoon of cheese over the omelet. Let rest until omelet has set; approximately 30 seconds.
FOLD the omelet on two sides using the rubber spatula, fold. Transfer to a serving plate and keep warm. Repeat with the remaining ingredients to make 2 omelets total.