In a saucepan, combine the cider, bourbon, brown sugar, corn syrup, vinegar, cinnamon stick and vanilla bean and seeds; bring to a boil. Simmer for 10 minutes, then add the cream. Cook over moderately high heat until reduced to about 1 cup, about 25 minutes. Transfer to a pitcher; let cool slightly. Discard the vanilla bean and cinnamon stick.
Preheat the oven to 350°. Butter and flour a 9-inch round cake pan. Line the bottom with parchment paper; butter and flour the paper. In a bowl, whisk the flour, salt, baking powder and baking soda. In a glass measuring cup, mix the apple cider with the canola oil and bourbon. In a large bowl, using an electric mixer, beat the butter until creamy. Add the brown sugar and vanilla seeds and beat until smooth. Add the eggs, 1 at a time, beating well between additions. Beat in the dry ingredients in 3 batches, alternating with the cider-bourbon mixture, until smooth.
Spread two-thirds of the batter in the prepared cake pan. Drizzle half of the cider sauce over the batter. Dollop the remaining batter on top, then spread the batter over the cider sauce, being careful not to mix them together. Bake the cake in the center of the oven for about 40 minutes, or until springy when touched. Let cool for 10 minutes in the pan, then turn the cake out onto a wire rack and let cool for 15 minutes. Peel off the parchment paper.
Rewarm the remaining cider sauce. Cut the cake into wedges and serve warm with vanilla ice cream, Pumpkin Seed Brittle and the cider sauce.
The cake and cider sauce can be refrigerated for up to 3 days.
Cooking Club Tip: Choose vanilla beans that are plump, moist and fragrant, not dried out and brittle. Beans from Madagascar (thought by many to be the best) and Mexico have a rich vanilla taste. To scrape out the seeds, open up the bean and run the blunt edge of the knife from one end to the other. The contents of one whole bean is equal to one tablespoon pure vanilla extract. To store vanilla beans, wrap them tightly in plastic wrap and put in an airtight container.