Ube Basque Cakes

Whisk together cake flour, all-purpose flour, baking powder, and salt in a medium bowl; set aside. Beat granulated sugar and butter with a stand mixer fitted with the paddle attachment on medium speed until smooth, about 3 minutes. Add eggs; beat on medium speed until mixture is light and fluffy, about 2 minutes, stopping to scrape down sides of bowl as needed. Beat in vanilla until incorporated, about 10 seconds. Add flour mixture all at once. Pulse mixer 3 to 4 times to begin incorporating flour mixture. Beat on medium-low speed until evenly combined, about 45 seconds.

Using a 1 1/2-inch scoop, drop 6 balls of batter (about 1 1/4 ounces or 2 1/2 tablespoons each) on a large piece of parchment paper, spacing balls at least 3 inches apart and away from outer edge of paper. Place a second piece of parchment paper on top of balls. Using bottom of an 8-cup liquid measuring cup, press to flatten each ball into a 4-inch round (about 1/8 inch thick). Repeat process with remaining batter and 6 additional sheets of parchment paper to form 24 dough rounds total. Stack the parchment-sandwiched rounds on a baking sheet. Freeze until very firm and parchment can be peeled off cleanly from batter, at least 2 hours or up to 3 days.

Stir together ube, cream, and granulated sugar in a small saucepan. Bring to a boil over medium-high. Reduce heat to medium-low; cover and simmer, stirring occasionally, until ube is very soft, 25 to 30 minutes. Using an immersion blender, process mixture in pan until very smooth, about 2 minutes. Let ube jam cool to room temperature, about 30 minutes.

Whisk together egg and egg yolk, brown sugar, granulated sugar, and cake flour in a medium-size heatproof bowl; set aside. Bring cream to a simmer in a small saucepan over medium. Reduce heat to medium-low. Whisking egg mixture constantly, gradually ladle about 2/3 cup of the hot cream into egg mixture. Pour egg mixture into remaining hot cream in pan. Cook over medium-low, whisking constantly, until mixture thickens to a pudding-like consistency, 2 to 3 minutes. Press pastry cream through a fine wire-mesh strainer set over a large bowl; discard solids. Stir in vanilla and salt. Press plastic wrap directly onto surface. Let cool to room temperature, about 30 minutes.

Whisk together cooled pastry cream and ube jam in a medium bowl. Ube pastry cream can be stored in an airtight container in refrigerator up to 3 days.

Preheat oven to 350°F with racks in upper third and lower third positions. Line 2 rimmed baking sheets with parchment paper. Lightly coat 12 (4- x 1-inch) ring molds with cooking spray; arrange 6 molds evenly spaced on each prepared baking sheet. Remove 1 sheet of batter rounds from freezer. Working quickly to avoid batter softening, peel off parchment paper, and carefully press 1 batter round into bottom of 1 mold. Repeat process with 5 additional batter rounds and ring molds. If batter round cracks, press the pieces back together in the mold. Repeat procedure with second sheet of batter rounds and remaining 6 ring molds.

Transfer ube pastry cream to a pastry bag with a 1-inch hole cut in the tip. Starting in the center and working outward, pipe about 2 tablespoons ube pastry cream in a spiral on each batter round, leaving a 3/4-inch border. Top ube pastry cream spirals evenly with huckleberries (a scant 1 tablespoon each). Remove 1 sheet of batter rounds from freezer. Working quickly, peel off parchment paper. Top each mold with a second batter round, pressing just until it touches the huckleberries. (The layers will fuse together in the oven.) Repeat process with remaining sheet of batter rounds and remaining molds.

Bake cakes in preheated oven until deep golden brown on top, 38 to 46 minutes, rotating baking sheets between top and bottom racks halfway through bake time. Transfer baking sheets to a wire rack; let cakes cool completely, about 30 minutes. Run an offset spatula around the edge of each mold, and remove cakes.

Garnish top of each cake with powdered sugar, using a 3-inch stencil of the Filipino sun, if desired.