Preheat the oven to 350 degrees F. Grease or line a 9x9” baking pan. Lining the pan with parchment paper that overhangs on at least 2 sides will allow easy removal of the bars. Set the pan aside.
In a large bowl, combine the all-purpose flour, almond flour, sugars, baking powder, and salt.
Add the softened butter. Combine the ingredients with a pastry blender or with your fingers. It should be crumbly, not a cohesive ball. Alternatively, you can pulse in a food processor with the blade attachment.
Place half of the flour mixture (approximately 2 cups) into the pan and firmly press (with fingers, spoon, or flat-bottomed glass) the mixture onto the bottom of the pan to create the base. Set aside.
Add the oats to the remaining flour mixture and combine by rubbing the oats into the butter with your fingers to keep the mixture’s crumbly texture. Set aside.
In a medium bowl, mix the sugar and cornstarch. Add the strawberries and toss. Drizzle the balsamic vinegar over the berries. Using a meat masher or fork, mash some of the berries. It’s not necessary to mash them all; you’re not making jam. Just adding some texture to them.
Spread the strawberry mixture evenly over the base. Be sure to get all the juices into the pan, too.
Sprinkle the remaining crumble evenly over the strawberries. Gently press the crumble mixture onto the strawberries. You want the crumble to adhere well, but not be submerged.
Bake in the preheated oven for 40 to 50 minutes, or until the mixture is slightly bubbling and lightly browned. (Mine was perfect in 45 minutes.)
Remove the bars from the oven and place the pan on a cooling rack. Allow the bars to almost completely cool in the pan. Run a sharp knife around the edges of the pan that do not have parchment. Lift the bars out by the parchment overhang, and allow them to completely cool before cutting into squares.
The bars are best served at room temperature. They can be kept refrigerated in an airtight container for up to 5 days.