Preheat the oven to 180 C (350 F), grease a 20 cm (8 inch) square baking pan, and line it with parchment paper leaving an overhang on the sides for easy lifting later.
In a large mixing bowl, combine the melted butter, flour, sugar, baking powder, and salt until the mixture looks crumbly with no dry patches, then add the egg and stir until everything comes together.
Press about two-thirds of this dough evenly into the bottom of the prepared pan, bake it for 15 minutes until lightly golden, then set it aside to cool for about 5 minutes.
While the base is baking, place the diced apples in a bowl and toss them with the flour, sugar, cinnamon, and nutmeg until every piece of apple is well coated.
Stir the rolled oats into the remaining dough, mixing with a spoon or your hands until combined, then break it into small clumps for the crumble topping.
Spread the apple filling evenly over the baked crust, sprinkle the crumble topping all over the apples, and bake for 30 to 35 minutes until the top is golden and crisp.
Let the bars cool completely in the pan, then use the parchment overhang to lift them out, cut into squares, and serve plain or with caramel sauce and ice cream.