Line bottom and sides of a 13- x 9-inch pan with heavy-duty aluminum foil, allowing 2 to 3 inches to extend over sides; lightly grease foil.
Pulse 2 cups flour, 1/2 cup powdered sugar, and 1 cup butter in a food processor 5 to 6 times or until mixture is crumbly. Press mixture onto bottom of prepared pan.
Bake at 350 on an oven rack one-third up from bottom of oven 25 minutes or just until golden brown.
Whisk together eggs and next 3 ingredients in a large bowl until blended.
Combine baking powder, salt, and remaining 1/4 cup flour; whisk into egg mixture until blended.
Pour lemon mixture into prepared crust.
Pulse 2 cups blackberries and 1/2 cup granulated sugar in a food processor 3 to 4 times or until blended.
Transfer mixture to a small saucepan. Cook over medium-low heat, stirring often, 5 to 6 minutes or until thoroughly heated.
Pour through a fine wire-mesh strainer into a bowl, gently pressing blackberry mixture with back of a spoon; discard solids.
Drizzle over lemon mixture in pan.
Bake at 350 on middle oven rack 30 to 35 minutes or until filling is set.
Let cool in pan on a wire rack 30 minutes. Lift from pan onto wire rack, using foil sides as handles, and let cool 30 minutes or until completely cool.
Remove foil, and cut into 24 (2-inch) squares; sprinkle with powdered sugar.
