Make the pineapple the day before you plan to serve the dessert.

Preheat oven to 375 degrees F.

Melt the butter over medium-high heat in a deep heavy ovenproof pot, large enough to hold the pineapple on its side. Add the brown sugar and vanilla bean. Cook, stirring, to melt and caramelize the sugar, 3 to 5 minutes. Lay the pineapple in the pot and let simmer, turning on all sides to brown. Add the water to the pan, cover and put pan into the oven. Baste occasionally, until roasted through to the core, about 1 hour.

If needed, add more water a tablespoon at a time. Let the pineapple cool in the pan. Then, lift out the pineapple, wrap, and refrigerate overnight. Reserve the pan juices.

Make the sherbet: Combine the sugar and water in a saucepan and bring to a boil. Let cool slightly, then whisk in the buttermilk and lime juice. Refrigerate until cold, then freeze in an ice cream machine.

When ready to serve, assemble the dessert: Thinly slice the pineapple (I use a meat slicer). If desired, re-warm the pan juices. Arrange slices overlapping on dessert plates. Place a scoop of sherbet in the center of the pineapple slices and drizzle the pan juices on top. Serve immediately.