1. Make the dough: Combine butter, granulated sugar, yolks, salt, vanilla, and flours in the bowl of a food processor. Process until dough comes together, 30 to 60 seconds. Turn dough out and divide in half. Shape each half into a 3-by-5-inch rectangle and wrap tightly in plastic wrap. Chill for 1 hour.

2. Make the filling: Heat figs and port in a small saucepan over medium until just simmering. Let stand until liquid is absorbed, about 10 minutes.

3. Transfer fig mixture to a food processor. Add vanilla bean seeds, brown sugar, and salt; process until a paste forms, 1 minute.

4. Preheat oven to 325 degrees with rack in the middle position. Roll out one piece of dough between lightly floured sheets of parchment paper into an 8-by-11-inch rectangle. Spread half of the filling lengthwise down the center of the dough to form a 2 1/2-by-9-inch rectangle. Use parchment to lift dough and fold over the filling to cover. Pinch edges to seal. Refrigerate on parchment until firm, 15 minutes. Repeat process with remaining dough and filling.

5. Remove both pieces of filled dough from refrigerator. Use parchment to carefully flip dough, seam-side down, onto a single parchment-lined baking sheet. Let stand 5 to 10 minutes. Combine egg and cream, and lightly brush dough. Sprinkle generously with turbinado sugar. Bake until golden brown, 30 to 35 minutes. Let cool on pan before slicing crosswise into 3/4-inch-thick pieces.