COOK and stir bacon in small nonstick skillet until crisp. Remove bacon from skillet with slotted spoon, reserving 1 tsp. drippings in skillet. Drain bacon on paper towels. Add kale to reserved drippings; cook on medium heat 5 to 6 min. or just until kale is wilted. Transfer to bowl; wipe skillet clean.
WHISK eggs, milk and mustard until blended. Melt butter in same skillet on medium heat. Add egg mixture; cook 5 to 6 min. or until almost set, occasionally lifting edge with spatula to allow uncooked portion to flow underneath. Top with cheese; cook until egg mixture is set but top is still slightly moist.
SPOON kale onto half the omelet; top with bacon and pepper. Slip spatula underneath omelet, tip skillet to loosen and gently fold omelet in half. Slide or flip omelet onto plate; cut in half.