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Prepare the crust: In a stand mixer fitted with paddle attachment, cream butter and sea salt on medium speed for about 1 minute. Scrape down sides of bowl and paddle with a rubber spatula and add confectioners’ sugar. Combine with butter at low speed. Once incorporated, scrape down bowl and paddle. Add almond flour and vanilla extract and combine at low speed.
Gradually add egg and a quarter of the flour (scant 1/2 cup or 55 grams). Beat at low speed until just incorporated. Scrape down bowl and paddle. Gradually add remaining flour and mix just until dough comes together, stopping from time to time to scrape in any mixture adhering to sides and bottom of bowl. Do not overbeat. Dough should be soft to the touch.
Separate dough into two equal portions. Gently press each portion into a 1/2-inch-thick rectangle. Double-wrap airtight in plastic wrap. Refrigerate one dough portion for at least 4 hours, preferably overnight; chill or freeze the second portion for another use.
Very lightly butter a 9-inch tart pan with a removable bottom. (You should not be able to see the butter.) On a lightly floured surface, roll out dough to a 10 1/2-inch circle, 1/4 inch thick. Dust work surface and dough often, and work quickly so dough remains cold. Gently roll dough over lightly dusted rolling pin and transfer to pan, gently easing it in and trimming the top edge. Chill uncovered for at least 1 hour, preferably longer.
Prepare the tart: Heat oven to 325 degrees. Sift together almond flour, confectioners’ sugar, cornstarch and flour into a medium bowl.
Place butter, salt and vanilla and almond extracts in a stand mixer fitted with the paddle and beat 1 minute at medium speed. Scrape down bowl and paddle, and add almond flour mixture. Beat at medium speed for 1 minute, until incorporated. Stop, scrape down bowl and paddle, then turn on machine and gradually add egg. Add rum and beat at medium speed until egg and rum are incorporated.
Remove tart shell from refrigerator and place on a baking sheet. Using a fork, pierce rows across surface of crust, about 1 inch apart. Scrape almond cream onto crust and, using a small offset or rubber spatula, spread evenly over crust.
Place in oven and bake 40 minutes, until crust and almond cream are golden brown and the tip of a knife comes out clean when inserted into cream. Remove from oven and let cool for 40 minutes on a rack.
Using a small spatula, spread fig jam over surface of tart in an even layer.
Remove stems from figs. Cut small and medium figs into quarters, large figs into sixths or eights. Arrange in concentric circles, starting with the rim, with the stem end down. Slices should angle upwards. If not serving right away, refrigerate. Dust with powdered sugar just before serving.