Start by preheating your oven to 400°F so it's nice and hot by the time your pie is ready to go in. Fit your pie crust into a 9-inch pie plate, pressing it gently into the edges and crimping the border however you like. Set it aside on your counter while you prepare the filling, so the crust doesn't dry out or crack.
Peel, core, and thinly slice your apples so they cook evenly and soften nicely in the oven. In a large bowl, toss the apple slices with the ½ cup of white sugar and the cinnamon until every slice is coated. I like to let them sit for a few minutes so the sugar starts to draw out the juices a little.
Arrange the sugared apple slices into your prepared pie crust, mounding them slightly in the center since they'll shrink a bit as they bake. Make sure to scrape in any extra cinnamon-sugar mixture left in the bowl, because that's pure flavor you don't want to waste. Press the apples down gently so they settle into an even layer.
In a separate bowl, whisk together the ⅓ cup sugar and the flour until well combined. Add your cold, cubed butter and use a pastry cutter or your fingertips to work it into the dry ingredients. Keep going until the mixture looks like coarse, pebbly crumbs rather than a smooth paste.
Sprinkle the crumble topping evenly over the apples, covering the entire surface so every bite gets some crunch. Place your pie on a baking sheet to catch any drips, then slide it into the oven. Bake for 35 to 40 minutes, until the apples are tender and the topping is a deep golden brown.
Once baked, pull the pie out and let it rest on a wire rack for at least an hour before slicing. I know it's tempting to dig right in, but cooling helps the filling set so your slices hold together instead of sliding apart. Serve warm or at room temperature, whichever you prefer.