Slice each fig lengthwise into 4 pieces (try not to rip skin; the nicest figs will be used to garnish cake). Place in a medium saucepan, add enough water to cover, and bring to a simmer over medium-high heat. Remove from heat, cover, and set aside until figs are soft, about 20 minutes.
Preheat oven to 350°. Line a 4" × 13" terrine mold with greased parchment paper. Sift flour, salt, baking soda, cinnamon, mace, and allspice together into a large bowl. Set aside.
Beat butter with an electric mixer, then gradually add brown sugar and beat until fluffy. Beat in eggs, one at a time, then add vanilla and sour cream. Slowly add dry ingredients, mixing until combined.
Using a slotted spoon, transfer three-fourths of the figs from poaching liquid to batter. Mix with a wooden spoon, then fold in almonds. Spoon into mold and bake until a toothpick inserted in center comes out clean, about 1 hour and 15 minutes. Cool for 30 minutes, then carefully turn out onto a wire rack and cool completely.
Meanwhile, remove remaining figs from poaching liquid and place in a bowl. Add sugar to poaching liquid and reduce over medium-high heat until thick and syrupy, about 15 minutes. Stir in armagnac, then pour over figs and set aside to macerate for at least 20 minutes.
Using a skewer, poke holes all the way through cake. Spoon syrup over cake and overlap remaining fig slices on top. (Store, wrapped in plastic wrap, in refrigerator for up to 1 week.)