How to Make It
Preheat the oven to 350°. Butter a 9-inch round springform pan and line the bottom with parchment paper. Butter the parchment. In a medium bowl, whisk the flour with the baking powder, baking soda and 1/4 teaspoon of salt. In the bowl of a stand mixer fitted with the paddle, beat the stick of butter with 1 cup of the sugar and the vanilla seeds at medium-high speed until light and fluffy, about 4 minutes. Beat in the eggs, one at a time. At low speed, alternately beat in the dry ingredients and buttermilk until just combined.
Scrape the batter into the prepared pan and bake for about 35 minutes, until a toothpick inserted in the center comes out clean. Let the cake cool on a wire rack for 30 minutes, then remove the ring and let cool completely.
Meanwhile, in a medium saucepan, combine the peach chunks with 1 cup of the sugar and a large pinch of salt. Bring to a boil, stirring until the sugar is dissolved, then simmer over moderate heat, stirring occasionally, until the fruit is broken down and the sugar is just beginning to caramelize, about 20 minutes. Let cool.
Using a serrated knife, trim the cake to form a 7-inch round, then trim the domed top to flatten it; reserve the scraps for another use. Discard the parchment. Cut the cake horizontally into 2 even layers and place them side by side on a baking sheet. Spread the peach jam evenly on top of both cakes. Freeze until the jam is firm, about 15 minutes.
Place a clean 9-inch springform pan on a rimmed baking sheet. Place one of the cake rounds in the center of the springform pan. Spread 2 1/2 pints of the ice cream over the jam and around the side of the cake, filling the space between the cake and the pan. Level the ice cream with an offset spatula. Freeze until the ice cream is firm, about 20 minutes.
Place the second cake layer on top of the first. Spread the remaining 2 1/2 pints of ice cream over the jam and around the side of the cake, as with the first layer. Level the ice cream with the offset spatula; the ice cream should be flush with the top of the springform pan. Cover and freeze overnight.
Rub a hot, wet towel around the outside of the pan to loosen the cake. Remove the ring and transfer the cake to a platter. Return the cake to the freezer.
In the bowl of a stand mixer fitted with the whisk, beat the heavy cream with the remaining 1 tablespoon of sugar until firm peaks form. Spread the whipped cream on top of the cake and arrange the peach wedges in a decorative pattern on top. Freeze the cake for 20 minutes before serving.