Cut the pineapple quarters crosswise into roughly ¼- to ½-inch-thick slices. Pile the slices into a bowl, cover it, and set aside. The fruit is at its best when served at room temperature. If you prepare it ahead and need to refrigerate it, take it out about 30 minutes to one hour before serving so that it may warm up.

To make the sauce, melt the butter in a 1-quart, non-reactive saucepan. Add the water and lemon juice and stir to mix.

Add the brown sugar and stir until it is dissolved.

Scrape the vanilla bean halves to remove the tiny seeds. Add the seeds and bean halves to the pan.

Bring the mixture to a boil over medium-high heat, stirring occasionally, lower the heat, and boil gently until thickened, about 7 minutes.

Remove the pan from the heat and let it sit for a couple of minutes until the syrup has stopped boiling.

Whisk in the cream until the mixture is smooth. Add the remaining tablespoon vanilla extract to taste, adding 1 teaspoon at a time, mixing well, tasting, and adding more as you like.

Remove the vanilla bean halves, scraping off the sauce, and set aside. (Rinse and dry the pods and save for another use.)

Pour the sauce into a bowl or pitcher. At this point, you may serve the sauce warm, or cool to room temperature.

When ready to serve, place the pineapple slices in individual bowls and pour the sauce over the fruit.