Line an 8-inch square baking pan with foil. Spray the foil with nonstick cooking spray and set aside. Place rack in middle of oven and preheat oven to 350 degrees.
In a large bowl, whisk together both flours, baking powder and salt. In the bowl of a stand mixer fitted with a paddle attachment, beat butter and sugar on medium speed until light and fluffy, about 3-6 minutes. Add the egg and vanilla and beat until combined. Add the flour mixture just until combined. Do not overmix. Set aside 3/4 cup of the dough for the topping.
Transfer the remaining dough to the prepared pan pressing it firmly into a nice even layer. You can use a greased spatula to do this. Bake until golden, about 15-20 minutes.
While the dough is baking, roll the reserved 3/4 dough between two sheets of parhcment paper that have been sprayed with nonstick cooking spray. Roll into an 8-inch square. Trim as needed. Transfer the dough in the parchment to a baking sheet to the freezer.
Spread the cooled fig mixture evenly over the crust in your pan. Lay the top crust over the filling, pressing lightly. Bake until the top crust is lightly golden brown, about 25-30 minutes, rotating the pan halfway through the baking time.
Remove the bars from the oven to a wire rack to cool, about 2 hours. Use the foil overhang to remove the bars from the pan. Cut into squares. Bars can be kept in an airtight container in the fridge.