Preheat oven to 350°. Using a large piece heavy-duty aluminum foil, line the bottom and sides of a 13- x 9-inch pan. Allow 2 to 3 inches to extend over sides; lightly grease foil.

In a food processor, pulse 2 cups flour, powdered sugar, and butter until mixture is crumbly, about 5 or 6 times. Press mixture into the bottom of the prepared pan.

Place the pan on an oven rack one-third up from bottom of oven. Bake the crust for 25 minutes or just until golden brown.

In a large bowl, whisk together eggs, sugar, lemon zest and lemon juice until blended.

In a small bowl, combine the remaining 1/4 cup flour, baking powder, and salt. Add to egg mixture and whisk until blended. Pour mixture evenly over prepared crust.

Add the blackberries and 1/2 cup white, granulated sugar to a food processor. Pulse 3 to 4 times or until blended.

Add the blackberry mixture to a small saucepan. Over medium-low heat, cook stirring frequently 5 to 6 minutes or until heated.

Place a fine, wire-meshed strainer over a medium bowl. Pour the blackberry mixture into the strianer. Using the back of a spoon, gently press the mixture against the wire mesh. Discard solids.

Drizzle the blackberry mixture over the lemon mixture in pan. Bake at 350° on middle oven rack 30 to 35 minutes or until filling is set.

Remove pan from oven and place on a wire rack to cool, 30 minutes. Using the overhanging foil sides as handles, lift the contents from the pan and place onto wire rack. Allow to cool completely, at least 30 minutes.