For the cake: Preheat the oven to 350 degrees F. Butter a 7 by 11-inch baking dish.
In a medium saucepan, bring the figs, cinnamon sticks and 2 cups water to a boil and then let simmer over medium heat until almost all the liquid has evaporated, about 10 minutes. Discard the cinnamon sticks. Set aside.
In a food processor, beat the softened butter with the brown sugar and vanilla extract until the mixture is creamy, about 2 minutes. Beat in the eggs, 1 at a time, scraping down the side of the bowl after each addition, and continue beating until the mixture is light, about 3 minutes. Add the fig mixture and process until pureed. Add the flour and baking soda. Pulse just until a smooth batter forms. Set aside.
Scrape the batter into the prepared baking dish and bake until the fig cake is springy and a toothpick inserted in the center comes out clean, 50 to 60 minutes.
For the toffee sauce: In a medium saucepan, melt the butter, add the brown sugar and continue cooking for 5 minutes. Add the heavy cream, bring to a boil, stirring once in a while over moderate heat until it starts to thicken, 8 to 10 minutes. Stir in the vanilla. Keep warm.
For the candied figs: In a small saucepan, bring the sugar and 1 cup water to a boil. Let it simmer until it reaches 300 degrees F on a candy thermometer, about 10 minutes. Turn off the heat. Place the figs, flesh-side down, on a tray lined with parchment paper. Pour the sugar mixture over the figs to coat them. Let cool at room temperature for 10 minutes until ready to serve.
Using a fork, poke holes all over the top of the warm fig cake. Drizzle half of the hot toffee sauce over the cake. Return the cake to the oven and bake for 5 minutes longer until the toffee sauce is bubbling around the edges but not fully absorbed.
Serve the pudding warm with the remaining toffee sauce, ice cream, if desired, and candied figs.