Preheat the oven to 350. Line a 9 inch square baking pan with parchment paper, or spray with non-stick baking spray.
In a small saucepan, melt the butter over medium heat. Continue to cook the butter, swirling the pan occasionally, until nutty brown solids form on the bottom of the pan. This should take about 5 minutes.
Meanwhile, in a bowl, combine 2 cups of the flour (reserving 1/4 cup of flour for later) with the brown sugar, salt and baking powder.
As soon as the butter is browned, use a spatula to scrape the bottom of the pan to scrape up all the browned bits. Pour the hot butter over the flour mixture. Stir with a spoon to form it into a soft dough.
Scoop 3/4 of the dough (reserving 1/4 of the dough for the crumb topping) into the prepared baking pan, and use your hands to press the dough evenly into the bottom of the pan. Bake for 15 minutes.
Take the reserved 1/4 cup of flour and add it to the remaining 1/4 dough. (You may not need to use the entire 1/4 cup of flour - I usually start with 2 tbsp, then add a little more if the topping seems too moist.) Use a fork or pastry cutter to work the flour into the dough until you have a crumbly topping. Set aside.
After the crust bakes for 15 minutes, take the pan out of the oven. Pour the apple filling over the hot crust, then sprinkle the apples with the crumb topping. Bake for another 30 minutes, until the filling is bubbling and the topping is golden brown.
Cool completely before cutting into squares.
Just like with apple pie, the apple filling in these bars will take a while (4-6 hours) to set and cool, before you can cut squares. If you don't mind them warm and messy, there's no need to wait that long, and you can just use a spoon to scoop up a serving to serve warm with ice cream.