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For the donut holes: In the bowl of a stand mixer, whisk together the milk and yeast by hand until completely dissolved, then whisk in the sugar. Whisk in the vanilla extract, followed by the eggs. Slowly add in the flour and salt.
Using the dough hook attachment, blend on low speed just until the dry ingredients get incorporated, about 30 seconds, then turn up to medium speed and add the butter in pieces.
Continue to mix until the dough comes together and pulls away from the sides, 5 to 6 minutes (the dough will be very sticky).
Flip the dough onto a well-floured work surface, and roll to an even 1-inch thickness. Using a 1-inch round cutter, cut out the donut holes. Transfer to a wire rack and place in a cold oven. Allow the donut holes to rise for 30 minutes.
Heat at least 2 inches of vegetable oil in a large Dutch oven until it registers 360 degrees F on a deep-frying thermometer. In batches, carefully fry the donut holes, turning frequently, for 2 to 3 minutes.
With a slotted metal spoon, transfer the donut holes to a cooling rack and allow to cool for 15 to 20 minutes.
For the pineapple glaze: Meanwhile, beat together 3 tablespoons of the butter, 2 ounces of the pineapple juice, 1/4 tablespoon of the vanilla paste and 1/2 teaspoon oil. Then beat in the confectioners' sugar until well combined. Beat in the remaining 3 tablespoons butter, 1 1/2 ounces pineapple juice, 1/4 tablespoon vanilla paste and 1/2 teaspoon oil. Microwave until just warmer than room temperature, 15 seconds.
For the pina colada donut holes: Place the warm pineapple glaze in a large bowl and the toasted coconut flakes in another bowl. Dip the donut holes in the glaze, then immediately toss them in the toasted coconut flakes.
Fill a 1-millimeter plastic pipette with the shooter filling and poke it into each donut hole to fill. Serve immediately.
Special equipment: a 1-inch round cutter; a deep-frying thermometer; a 1-millimeter plastic pipette