Step 1

In a medium bowl, combine the flour and salt. In a mixing bowl, combine the yeast and milk, whisking thoroughly. The combination should become frothy; this means that the yeast is activating. Set aside for a few minutes.

Step 2

In another bowl, whisk together the granulated sugar, extract and eggs. Whisk the yeasted milk into the eggs and combine thoroughly. In a stand mixer fitted with the paddle attachment, pour in the combined liquids. Pour the flour and salt mixture on top of the liquids. Mix on low speed for 1 minute. Change from the paddle to the dough hook and continue to work the dough on medium-low speed. Begin adding the butter, one cube at a time, until each cube of butter is thoroughly incorporated. Allow the dough to mix until it forms a ball and the dough cleans the sides of the bowl, 20 to 30 minutes. Grease a medium bowl with the olive oil, then cover and allow to rest in the refrigerator overnight.

Step 3

The next day, pull the dough from the refrigerator and knead into a smooth ball on a wood surface. The dough may become tacky; avoid adding any more flour. Return the dough to the bowl and let the dough rest again in the refrigerator for 5 to 10 minutes. Turn the dough out onto a heavily floured surface, then add a good amount of flour to the top of the dough. With a rolling pin, roll the dough to about 1/2-inch-thick. Use a 3-inch round cutter to punch out the dough. Place the donuts on lined sheet trays and cover. Place the doughnuts in the refrigerator and allow to rest for 12 hours.

Step 4

Pull the donuts from the refrigerator. Allow to proof while loosely covered at room temperature until the dough bounces back immediately after touching and is no longer tacky, about 1 hour.

Step 5

Meanwhile, set the canola oil in a deep-fryer to 350 degrees F. Line a sheet tray with parchment paper and set a rack on top.

Step 6

Place the doughnuts in the fryer basket and lower into the hot oil. Count to 20, then flip the doughnuts with chopsticks. Continue to cook, turning the dough every 2 to 3 minutes, until deep amber, 5 to 6 minutes. Remove from the fryer and place on the prepared rack. Immediately pour the Rum Glaze over the piping hot donuts. Let cool completely.

Step 7

Make holes in the tops of the donuts and pipe about 1 tablespoon Caramelized Pineapple Jam into each. Using a piping bag, pipe 2 to 3 ounces Pastry Cream into each. Using a piping bag with a fluted tip, pipe a dollop or rosette onto the tops of the donuts, covering the holes. Pipe a dollop of caramelized jam on top of each rosette and serve.

Step 8

Place the cardamom, cloves and cinnamon stick in a dry pan and gently toast over medium-low heat to release the aromas. Add the rums, sugars and vanilla seeds and bean. Allow to come to a boil. Using a propane torch, apply a flame and step away, allowing the alcohol to burn off. (Make sure to do this in a well-ventilated space.) Add the citrus zests and allow to boil for a few more seconds. Add the apricot jam and bring back up to a boil. Remove from the heat and strain out the solids. Allow to cool.

Step 9

Press the pineapple into a measuring cup, smashing it a little to get it leveled. Place the pineapple into a small pot. Add the granulated sugar and toss with the pineapple. Crush the pineapple and sugar together to help release the juices and begin melting the sugar. This helps prevent scorching.

Step 10

Add the lemon or lime juice and cook on low heat. (This will allow more of the fruit to break down.) Bring to a simmer, then increase the heat and cook until the sugar is a rich amber color and the mixture reaches 360 degrees F. Remove from the heat, then pour the pineapple and caramel into a blender. Place the lid on the blender and begin to puree; add the butter and continue to puree until smooth. Adjust the salt content if desired.

Step 11

Add the heavy cream and 1 1/2 teaspoons vanilla paste to a chilled glass or metal bowl and whip until soft peaks, 2 to 3 minutes. Set aside.

Step 12

Place the milk, granulated sugar, salt and remaining tablespoon and teaspoon vanilla paste in a saucepan over medium-low heat. Place the cornstarch and yolks in a medium bowl and whisk well. Set aside. Once the milk reaches 180 degrees F, pour over the yolk mixture and whisk briskly and thoroughly. Pour the mixture back into the pan and continue whisking until the pastry cream returns to 180 degrees F. Remove from the heat and quickly whisk in the butter and gelatin, if using. Strain the pastry cream through a chinois or fine-mesh strainer to remove anything that may affect the texture. Place the caramel wafers in the warm pastry cream, if using, and mix thoroughly. Place over an ice bath and let cool down, whisking, until the mixture reaches 65 degrees F. Fold in the whipped cream.