Coat a 9" x 13" baking pan with cooking spray. Line with parchment paper, leaving about 1" overhanging the long sides. Coat the parchment and short sides with cooking spray.
Filling
To prepare the filling: Combine strawberries, ⅓ cup sugar, cornstarch, lemon juice, 1 teaspoon vanilla, and a pinch of salt in a medium saucepan.
Cook over medium heat, stirring frequently, until the berries begin to break down and the mixture is thickened, about 10 minutes. Remove from heat.
Crust
To prepare the crust: Combine the flour, sugar, baking powder, and salt in a food processor; pulse until combined.
Add butter; pulse until well incorporated.
Lightly beat egg, oil, ice water, and vanilla in a small bowl.
With the motor running, add the mixture to the food processor. Pulse, scraping down the sides if necessary, until it begins to clump together, 30-45 seconds (it will look crumbly).
Measure out 1 cup for the topping.
Press the remaining mixture into the prepared baking pan to form a bottom crust.
Pour the strawberry mixture over the crust, spreading evenly.
Sprinkle the reserved crust mixture on top.
Bake until the top is lightly browned, about 40 minutes. Transfer the pan to a wire rack and let cool for 15 minutes.
Use the parchment to lift out the bars and transfer to a cutting board. Cut into bars. Let cool completely before serving, about 25 minutes more. Refrigerate for up to 3 days.