Prep. Preheat the oven to 350˚F. Lightly grease an 8x8 baking dish with cooking spray and set aside.
Make the crust. In a food processor, combine the all purpose flour and whole wheat flour with the powdered sugar, walnuts, and salt; pulse to combine. Add butter and oil; continue to pulse until the mixture is crumbly. Add cold water and pulse until mixture resembles coarse meal.
Bake. Transfer the crust mixture to the baking dish and press it into the bottom of the pan. Pop it in the oven and bake for 18 minutes.
For The Filling
Make the filling. Meanwhile, in a mixing bowl, whisk together the sugar, 2 tablespoons all-purpose flour, lemon juice, lemon zest, eggs, and one egg white until thoroughly incorporated and smooth.
Lower the oven temperature. Remove the baked crust from the oven and let it stand for 2 minutes. Reduce oven temperature to 325˚F.
Assemble and bake. Arrange the fresh raspberries over the baked crust. Pour the lemon filling over the raspberries. Bake for 22 to 25 minutes or until the top is set.
Cool and chill. Remove the bars from the oven and let them cool completely in the pan set on a wire rack. Then, cover and chill them in the freezer for at least 1 hour.
Slice and serve. Place the bottom of the baking dish in a larger pan filled with hot water so it's easier to cut out the squares. Keep in the hot water for about 30 seconds. Sprinkle the top with powdered sugar. Slice into squares and serve.