Preheat the oven to 375 degrees F.
Place parchment paper on the bottom of the 8x8 pan that hangs over the side of the pan. (This is to make it easier to remove the Strawberry Crumb Bars after cooking.)
In a medium bowl, stir together 1/2 cup white sugar, flour, baking powder and salt.
Using a fork or a pastry cutter, cut the butter into the flour mixture. (VERY IMPORTANT: Make sure the butter is very cold.) Work with the dough until the pieces are very small. Dough will be crumbly. You can also pulse in a food processor to until you get pea sized crumbs.
In a separate measuring cup or small bowl, beat an egg with a fork. Add the egg and vanilla to the flour mixture, and combine until you have a crumbly mixture.
Pat half of the dough evenly into the bottom of the prepared pan.
In another bowl, stir together 1/3 cup sugar and cornstarch. Stir in the strawberries.
Spoon the strawberry mixture evenly over the bottom crust.
Crumble the remaining dough over the strawberry filling layer.
Bake in a preheated oven for 45 minutes, or until top is slightly golden brown. (My oven runs hot and baked in about 38 minutes.)
Cool completely before cutting into squares. I recommend refrigerating until ready to serve.