Like a fancy Fig Newton, this cake balances honey-sweetened jammy figs with toasty, bitter rye flour in a simple two-layer cake covered in a thin layer of brown butter frosting. Some of the jam — pureed with lots of lemon juice — gets poured over the top of the cake to add a hit of floral tartness. Turkish or Black Mission figs make the best filling for this cake, but use your favorite type; just make sure the figs are so ripe they can barely hold themselves together.
Make the cake: Heat the oven to 325 degrees. Grease the bottom and side of a round 8-inch light metal cake pan and dust with all-purpose flour, tapping out the excess. In a small bowl, whisk together both flours, the baking powder and salt.
In the bowl of a stand mixer or large bowl, combine the butter and sugar and beat on medium speed of a mixer until light and fluffy, about 3 minutes. Add the eggs one at a time, beating for 15 seconds between each addition. With the mixer on low speed, add half the flour, followed by the milk and vanilla, and then the remaining flour, stopping when the batter just comes together. Use a rubber spatula to scrape down the bottom and sides of the bowl to make sure the batter is evenly mixed.
Scrape the batter into the prepared pan and smooth the top. Bake until the cake is golden brown on top and a toothpick inserted in the center comes out clean, about 40 minutes. Remove the pan from the oven and let cool for 10 minutes. Unmold the cake and place it right side up on a rack to cool completely.
While the cake bakes, make the fig filling and sauce: Cut the figs into 1/2-inch pieces then place in a medium saucepan, along with the water, honey and salt and bring to a boil. Reduce the heat to medium and cook, stirring occasionally, until the figs are soft and jammy and no loose liquid remains, 12 to 14 minutes. Remove the pan from the heat, stir in the vanilla and let the fig filling cool for about 10 minutes. Scoop 1/3 cup of the filling into a small blender or food processor, add the lemon juice and puree until smooth. Reserve the filling in the pan and the puree in the blender.
While the cake and filling cool, make the frosting: In a small saucepan or skillet, melt the butter over medium heat. Continue cooking, stirring occasionally, until the milk solids turn brown and smell nutty, 5 to 7 minutes. Remove the pan from the heat and scrape the brown butter into the bowl of a stand mixer or large bowl. Let the butter cool until solidified again. You can place the bowl in the refrigerator to speed this process along, but don’t let it get too firm; you want the brown butter solid but spreadable.
To the brown butter, add the powdered sugar, vanilla and salt and beat on low speed until smooth. While mixing, slowly drizzle in the milk, then increase the mixer speed to medium-high and beat until the frosting is light and fluffy, about 1 minute.
Using a long serrated knife, trim the top of the cake so it is flat and discard the trimmings. Cut the cake through its equator into two disks; set the top disk to the side. Transfer the bottom disk to a cake stand or platter, then scrape the fig filling from the pan over the top, smoothing it evenly and leaving a ½-inch border around the edge. Flip the second cake disk upside-down and place over the filling, pressing gently to spread the filling just to the edges and ensure that the cake is flat.
Scrape the frosting on top of the cake and use a small offset spatula or table knife to spread it evenly over the top and sides. Using the spatula or a spoon, lightly press around the top of the frosting, ½ inch inset from the edge, all over the top of the cake to create a depression. Pour the pureed fig sauce into the center of the cake and spread it evenly to fill the depression. Refrigerate the cake for at least 1 hour to firm the frosting before serving. Serve at room temperature and store any leftovers in the refrigerator.
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