Make the crust: Preheat the oven to 275 degrees. Butter a 9 x 9 inch cake pan. Cut a piece of parchment paper to fit the inside of the pan, with enough overhang so that the parchment covers the bottom and comes up and over the sides by a couple of inches creating a sling. Place the parchment inside the pan. Butter the bottom and up the sides of the parchment just a couple of inches.
Combine flours, brown sugar and salt together in a work bowl of a food processor. Process until combined. Add melted butter and vanilla. Pulse until fully combined and the texture of wet sand.
Transfer the dough to the prepared pan. Set the empty work bowl aside to use for the crumble (no need to rinse). Press dough evenly into the bottom of the prepared pan, using the bottom of a metal measuring cup to smooth it out and press it flat. Place the crust in the oven. Bake for 50-55 minutes or until golden brown and slightly firm to the touch.
Meanwhile, make the crumble: Combine all ingredients except the butter in the work bowl of a food processor. Pulse until combined and oats are slightly broken up, approximately 5-7 pulses. Transfer mixture to a bowl and add melted butter. Stir with a fork to combine. Set aside.
When the crust comes out of the oven, increase the oven temperature to 350 degrees. Spread the Apple Compote evenly over the top of the crust (a small offset spatula works really well here), taking care not to touch the hot pan.
Squeeze some of the crumble mixture together in your hand. Tear off irregular-sized pieces of the crumble and scatter it in an even layer over the top of the apple filling. Crumble pieces should be ¾ inch in diameter or smaller. Repeat with remaining crumble. Press gently on the top of the crumble with the back of a spatula to create an even layer and help the crumble adhere together.
Return the pan to the oven and bake for 50 minutes or until crumble topping is crisp and golden brown. Transfer pan to a wire rack to cool slightly before placing in the fridge for a few hours to chill bars before slicing.
Before serving, run a small knife along the inside edge of the pan. If you have chilled the bars in the fridge, gently lift the parchment paper sling and remove bars in one piece from the pan. Cut into squares and serve. If you have simply let the bars cool to room temperature without chilling, slice the bars right in the pan and serve.