Peel and dice the apples. Place them in a pan with butter, half the sugar, cinnamon, lemon zest, and lemon juice. Cook over medium-high heat, stirring occasionally, until soft and caramelized. Separate egg yolks from the whites. In a bowl, mix the yolks with flour and milk, alternating gradually, to create a smooth batter. In a separate bowl, whisk the egg whites with the remaining sugar until stiff peaks form.
Gently fold the yolk mixture into the whipped egg whites, keeping the batter airy. Pour the mixture over the cooked apples in the pan. Cook until the bottom sets, then carefully flip to cook the other side.
Slice the omelette and serve warm, dusted with powdered sugar.
