In the work bowl of a food processor, place figs and 1 teaspoon (4 grams) vanilla. With processor running, add ½ cup (120 grams) water in a slow, steady stream; process until combined, 3 to 5 minutes.
Preheat oven to 350°F (180°C). Spray a 15¼x10¼-inch rimmed baking sheet with baking spray with flour. Line bottom of pan with parchment paper.
In a medium bowl, whisk together flour, cinnamon, baking powder, and salt.
In the bowl of a stand mixer fitted with the whisk attachment, beat granulated sugar and eggs at medium-high speed until pale and thick, about 4 minutes. Beat in remaining 1 teaspoon (4 grams) vanilla. With mixer on medium-low speed, add fig purée, beating until just combined. Using a rubber spatula, fold in flour mixture until just combined, being careful to knock out as little air as possible. Pour batter into prepared pan, spreading with an offset spatula.
Bake until lightly golden and cake springs back when lightly pressed in center, 14 to 16 minutes. Immediately run a thin knife or a small offset spatula around edges of cake. Using a fine-mesh sieve, generously dust top of cake with confectioners’ sugar. Place a clean kitchen towel on top of cake. Invert cake onto a baking sheet or a large cutting board; discard parchment. Starting with one short side, immediately roll up cake and towel together. Place, seam side down, on a wire rack, and let cool completely.
Carefully unroll cooled cake. Using a pastry brush, brush cake with Brown Sugar Simple Syrup. (There may be some syrup left over; don’t oversaturate the cake.) Spread 1½ cups (450 grams) Spiced Buttercream on cake, leaving a ½-inch border on short side that is bottom seam. Starting from opposite short side, gently reroll cake without towel. Place roulade, seam side down, on a wire rack or a baking sheet; refrigerate until set, at least 1 hour.
Trim about ½ inch from exposed sides of roulade. Spread remaining ½ cup (150 grams) Spiced Buttercream on top of roulade, and garnish with dried golden figs, if desired. Refrigerate in an airtight container for up to 3 days.