Adjust oven rack to middle position and heat oven to 350 degrees. Grease and flour eight 6-ounce ramekins, then line the bottom of each with a round of parchment paper. Place a kitchen towel on the bottom of a large roasting pan and arrange the ramekins on the towel. Bring a kettle of water to boil.
Whisk the flour, baking powder, and the salt together in a medium bowl. Combine ¾ cup figs with the warm water and baking soda in a glass-measuring cup (the figs should be completely submerged) and set aside to soften, about 5 minutes. Drain the softened figs and reserve the soaking liquid.
Pulse the remaining ½ cup figs and brown sugar in food processor until just combined, about five pulses. Add the reserved figs, soaking liquid, eggs, and vanilla and process until smooth, about 5 seconds. With the food processor running, pour melted butter through feed tube in steady stream. Transfer this mixture to a medium bowl and stir in softened figs. Gently stir in the flour mixture until just combined. Divided the batter evenly among the prepared ramekins.
Place the roasting pan in the oven and carefully pour enough boiling water into the pan to reach halfway up the sides of the ramekins. Cover the pan tightly with aluminum foil, crimping the edges to seal. Bake the cakes until puffed and small holes appear on their surface, 35 to 40 minutes.