In a medium bowl, whisk together all-purpose flour, almond flour, baking powder, and 3/4 teaspoon salt until thoroughly combined. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, combine butter and sugar and mix on medium speed until light and fluffy, about 3 minutes. Scrape down bowl and paddle, add the 1 large egg, and mix on medium speed until fully incorporated, about 1 minute. Scrape down bowl and paddle, add half of flour mixture, then mix on low speed until just combined, about 15 seconds. Repeat with remaining flour mixture, scraping down bowl and paddle as needed.
Scrape dough, along with any loose bits, onto a lightly floured work surface and, using your hands, bring dough together into a smooth ball (dough should be soft and slightly sticky but refrain from adding more flour). Divide dough into 2 equal portions (about 340g each) and shape into flat, round disks. Wrap each disk tightly in plastic wrap and refrigerate for at least 3 hours and up to 1 day.
Adjust oven rack to middle position and preheat oven to 350°F (180°C). Grease an 8-inch cake pan with butter and set aside. Remove dough disks from refrigerator and set on clean work surface.
In a medium bowl, whisk together pastry cream, rum (if using), and almond extract until thoroughly combined. Set aside.
In a small bowl, whisk together the remaining whole egg and egg yolk with the cream and the remaining 1/8 teaspoon salt until homogeneous. Set egg wash aside.
Place one dough disk on a lightly floured work surface. Using a rolling pin and adding flour as needed underneath and on top of the dough to prevent sticking, roll out to a 9-inch circle that’s about 1/4-inch thick. Transfer to the prepared cake pan by carefully rolling dough around the rolling pin, then unrolling over the pan. Press dough gently yet firmly into the corners and up the sides of the pan. Dust off excess flour with a pastry brush.
Add cherry jam to the center of the dough, and using the back of a spoon or a small offset spatula, spread in a smooth, even layer, leaving 1/2-inch of space around the perimeter. Repeat with prepared pastry cream, spreading evenly on top of jam.
Place second dough disk on a lightly floured work surface. Using a rolling pin and adding flour as needed underneath and on top of the dough to prevent sticking, roll out to a 8.5-inch circle that’s about 1/4-inch thick. Transfer to the cake pan by carefully rolling dough around the rolling pin, then unrolling over the filling. Using your hands, tuck the top layer of dough in along the space around the perimeter, making sure that the outermost edge of the dough is upright. Using your thumbs, seal the top and bottom edges of dough together. Fold the excess dough up and back over the top, pressing down around the perimeter to smooth out the dough. Dust off excess flour with a pastry brush.
Using a pastry brush, brush top surface all over with egg wash. Using the tines of a fork, gently scrape a decorative pattern onto the surface (a crosshatch pattern is traditional but you can be as creative as you like).
Bake cake until puffed and deep golden brown, about 45 minutes.
Transfer cake pan to a wire rack and let cool completely to room temperature, about 2 hours. Run a butter knife along edges to loosen, then invert onto wire rack, remove pan, and place cake right side up on a serving platter. Slice and serve.
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