1. Place 1/3 cup granulated sugar and butter in a large bowl; beat with a mixer until combined (about 1 minute). Add egg yolk, beating well. Stir in vanilla.
2. Place almonds in a food processor; pulse 10 times or until finely ground. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine nuts, flour, baking powder, and salt, stirring well with a whisk. Gradually add nut mixture to butter mixture, beating at low speed just until a soft dough forms, adding 3 tablespoons ice water, as necessary. Turn dough out onto a lightly floured surface; knead lightly 6 times or until smooth. Divide dough into 2 equal portions; wrap each portion in plastic wrap. Chill 1 hour or until firm.
3. Preheat oven to 375°.
4. Combine the remaining 2/3 cup granulated sugar, blackberries, cornstarch, and lemon juice; toss gently. Arrange berry mixture in a 13 x 9-inch glass or ceramic baking dish coated with cooking spray.
5. Unwrap dough. Roll each dough portion into a 13 x 9-inch rectangle on a lightly floured surface. Cut one rectangle, crosswise, into (1-inch-wide) strips. Cut remaining rectangle, lengthwise, into (1-inch-wide) strips. Arrange strips in a lattice pattern over fruit mixture; sprinkle dough with turbinado sugar. Bake at 375° for 50 minutes or until golden. Let stand 10 minutes.