In a medium bowl, whisk together both sugars, flour and cinnamon. Add the butter pieces and use your fingers to rub the mixture together until the butter is incorporated and the mixture resembles coarse crumbs.
Cover with plastic wrap and place in the refrigerator to chill while you prepare the apple filling.
Place the sliced apples in a baking dish. Squeeze juice of half a lemon over the apple and toss it together.
Sprinkle crumb topping over the apples.
Bake at 350 degrees for 35 to 40 minutes, until topping is nicely browned and apples are done.
Transfer to a wire rack to cool for 10 minutes. Serve warm, with a scoop of vanilla ice cream