Whisk eggs and milk together in a small bowl. Spray a medium non-stick skillet with cooking spray. Preheat to medium.
Pour egg mixture into skillet. Add chicken, onions and banana peppers to skillet. As eggs set at the edges, with spatula, gently lift cooked portion to allow uncooked egg to flow underneath. Cook until almost set, about 8 minutes.
Sprinkle cayenne and cheese on half of the omelette. Fold other half over cheese. Cut omelette in half and serve with nacho chips and sour cream.
Note: This is a delicate omelette, with a higher than usual proportion of milk, so it may be a bit difficult to fold. As an alternative, you can set the skillet under the broiler for a minute, to set the top, and serve it in wedges like a frittata.