To make the vanilla ice cream: Split the vanilla bean pods lengthwise and scrape out all the seeds. In a large saucepan, combine the milk and cream, add the seeds and pods, and bring to a slow boil over medium heat.
In a large bowl, using and electric mixer on high speed, beat the egg yolks and sugar until light and fluffy, about 2-3 minutes. Pour in half the boiling milk mixture and combine. Pour into pan and cook, stirring, over medium heat until thick enough to coat the back of a spoon.
Remove from heat and immediately strain into another bowl. Place over a second large bowl filled with ice and let cool completely, stirring every 30 minutes. When cool, freeze in an ice cream maker according to the manufacturer's directions.
To make the vanilla bean simple syrup: Bring sugar, leftover vanilla bean pods and water to a boil in a small saucepan. When sugar is dissolved, remove from heat. Remove vanilla bean pods. Cool syrup to room temperature.
To make the pineapple carpaccio: Cut off the top of the pineapple and, following the shape of the fruit, slice off the skin with a serrated knife, making sure to remove all the eyes. Using a serrated knife, slice the fruit into the thinnest possible rings.
Place each slice on a sheet of plastic wrap, brush with the syrup, rub the brown sugar over the surface, and top with a second sheet of plastic wrap. Repeat with the remaining slices, transfer to a rimmed backing sheet, and refrigerate overnight.
To make the banana beignets: In a medium bowl, combine the flour, salt, sugar and baking powder. Add the eggs and 1 cup water and whisk until a smooth paste forms. Refrigerate for at least 1 hour.
In a small bowl, combine the sugar and cinnamon; set aside.
In a deep fryer, heat the oil to 350 degrees. Working in batches, dip the bananas in the chilled batter and coat completely. Slip them into the oil and fry for about 2 minutes or until golden brown. Transfer to paper towels to drain. Dust with cinnamon-sugar mixture.
To assemble the dish: Arrange equal portions of the carpaccio in an overlapping circle on a dessert plate. Lay two pieces of fried banana in the center, and top with one large scoop of vanilla ice cream. Drizzle with vanilla bean syrup.