Prepare the ice cream layer: Line a 9-inch round cake pan with plastic wrap. Spread the ice cream in the pan using an offset spatula to create a smooth, even layer. Freeze until very firm, about 2 hours.
Make the pineapple topping: Preheat the oven to 350 degrees F. Butter the sides of another 9-inch round cake pan. Stir the granulated sugar and water in a medium saucepan over medium heat until the sugar dissolves. Cook, gently swirling the pan but not stirring, until the mixture is light brown, 6 to 10 minutes. Remove the pan from the heat and carefully whisk in the butter. Pour the caramel into the prepared cake pan, quickly swirling to coat the bottom before the caramel hardens (use a rubber spatula if needed to push the caramel to the edges). Lay the whole pineapple slice in the center of the caramel, then lay the remaining half slices around the whole slice, all facing the same direction. Place 1 maraschino cherry half cut-side up in the center of each slice.
Make the cake: Whisk the flour, baking powder, baking soda and salt in a large bowl. Whisk the milk, sour cream and vanilla bean paste in a liquid measuring cup until smooth. Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 4 minutes. Beat in the eggs, one at a time, until well combined. Reduce the speed to medium low and beat in the flour mixture in three additions, alternating with the milk mixture, until combined. Scrape the bowl with a rubber spatula, then increase the speed to medium high and beat 30 more seconds.
Pour the batter over the pineapple topping in the pan and spread in an even layer. Bake until well browned and a toothpick inserted into the center comes out clean, 55 minutes to 1 hour. Run a knife around the cake to loosen, then transfer to a rack and let the cake cool in the pan, 30 minutes. Invert the cake onto the rack to cool completely.
Meanwhile, make the pineapple compote: Combine the dried pineapple, pineapple juice, brown sugar, lemon juice, water and salt in a saucepan. Bring to a simmer over medium heat and cook until the pineapple is soft and the liquid is syrupy, 8 to 12 minutes. Remove from the heat and stir in the vanilla bean paste. Transfer the mixture to a food processor and pulse until the pineapple is finely chopped. Let cool.
Assemble the cake: Clean the 9-inch round cake pan and line with plastic wrap, leaving a 2-inch overhang all around. Slice the cake in half horizontally with a long-serrated knife. Place the half without the pineapple cut-side up in the pan. Top with the pineapple compote and spread evenly. Unwrap the frozen ice cream disk and carefully place on top of the compote. Place the remaining cake half, pineapple-side up, on top of the ice cream. Cover with plastic wrap and freeze until firm, at least 4 hours or overnight. Remove the cake from the pan, using the plastic wrap to help; transfer to a platter. Let the cake sit at room temperature for 10 minutes before slicing.