Make the cakes Position a rack in the center of the oven and heat the oven to 350˚F. Butter 2 10-inch round cake pans and line the bottoms with parchment. Butter the parchment and set the pans aside.
Combine 1-1/2 cups of the sugar with 6 Tbs. of water in a 2-quart, heavy-duty saucepan over medium heat. Stir gently until the sugar dissolves and the syrup comes to a boil, about 2 minutes. If there are any crystals on the sides of the pan, wash them down with a brush dipped in water. Once the syrup boils, stop stirring, and boil undisturbed until it turns a deep amber color, 10 to 20 minutes.
Immediately remove the pan from the heat and, without stirring or scraping, drizzle the syrup evenly over the bottoms of the cake pans. If there are any sugar crystals visible, pour the syrup into the pans through a sieve. It’s fine if the caramel isn’t evenly spread in the pans when it hardens – it will melt and spread as the cakes bake. Arrange the pear slices in a spiral pattern on top of the caramel in the cake pans, fitting them tightly together. Whisk the flour, baking powder, baking soda, and salt in a medium mixing bowl.
Halve the vanilla bean lengthwise and scrape the seed pulp from the inside. Mash the pulp with the flat of your knife and chop through it briefly to make sure you have no large bits. Remove the seeds from the cardamom pods and grind them with a mortar and pestle to a coarse powder.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large mixing bowl with a hand-held electric mixer), beat the butter and the remaining 2 cups sugar on medium speed until fluffy, about 5 minutes. Beat in the vanilla bean pulp, cardamom, and almond extract, scraping down the sides of the bowl as necessary. On low speed, beat in the eggs one at a time until fully incorporated, about 30 seconds per egg. Alternately add the flour mixture in 3 additions and the buttermilk in 2 additions, mixing just until the batter is smooth and falls slowly in a thick ribbon from a spatula.
Spoon the batter over the pears, carefully spreading it evenly out to the edges of the pans without disturbing the pears. Tap the pans on the counter a few times to settle the batter. Bake until a toothpick inserted in the center of each cake comes out clean, 45 to 55 minutes.
Let the cakes cool on racks until they are barely warm. Run a knife between the cakes and the pan sides to loosen them. If you encounter a sticky place, dip the knife in water and continue until the cake is free of the pan sides. Invert the cakes onto serving plates. Lift the pans off the cakes and peel off the parchment. If a bit of caramel stays behind anywhere on the edge of your pan, use the tip of a knife to lift it and put it where it belongs.
When completely cool, wrap the cakes tightly with plastic and store them at cool room temperature for at least several hours, and preferably overnight. As they rest, the moisture in the cakes equalizes and the caramel on the edges softens, making the cakes very easy to slice.
Serve the cakes When ready to serve, combine 1/3 cup of the sugar with 1/3 cup water in a small saucepan. Bring to a boil over medium heat, stirring frequently. When the sugar is completely dissolved, remove from the heat and stir in 2 Tbs. of the Cognac. Let cool briefly, then brush the syrup over the cakes to shine the pears.
Whip the cream, the remaining 1-1/2 Tbs. Cognac, and remaining 1-1/2 Tbs. sugar in a stand mixer fitted with the whisk attachment (or a hand-held electric mixer and a cold bowl) on medium-high speed to soft peaks, about 2 minutes. Slice the cakes and serve with the whipped cream.
