In a small bowl stir together butter and shortening. Cover; freeze 15 minutes. Meanwhile, in a large bowl stir together the flours, brown sugar, baking powder, and salt. Add butter mixture. Using your fingers, rub into flour mixture until coarse and pebbly.
In a 1-cup glass measure whisk together the 1/2 cup cream, egg yolk, and vanilla. Add to flour mixture. Stir with a fork just until combined. Stir in almonds. Gently knead until smooth, about 1 minute (dough should hold its shape without falling apart). Transfer to a lightly floured surface. Pat into a 6x6-inch square. Cut into 6 rectangles. Transfer to a parchment-lined baking sheet. Freeze 15 minutes. Preheat oven to 400º.
In a medium bowl stir together apricots, granulated sugar, and lemon juice; set aside.
In a small bowl, whisk together egg white and remaining 1 Tbsp. cream. Brush over tops of dough squares; sprinkle with turbinado sugar. Bake 12 to 15 minutes or until golden brown and a toothpick inserted near center comes out clean. Let cool on baking pan 2 minutes. Remove; cool completely on a wire rack. Serve with Mascarpone-Honey Cream and apricot mixture.
MASCARPONE-HONEY CREAM: In a large chilled mixing bowl beat 1 cup mascarpone cheese, 3 Tbsp. whipping cream, and 2 Tbsp. honey with an electric mixer on medium speed until fluffy.