In a small bowl, stir collectively heat milk, yeast, and 1 teaspoon (4 grams) sugar till nicely mixed. Cowl and let stand till foamy, about 10 minutes.
Within the bowl of a stand mixer, whisk collectively 2 cups (250 grams) flour, salt, and remaining ½ cup (100 grams) sugar by hand. Add yeast combination; utilizing the paddle attachment, beat at low velocity till mixed. Add egg, egg yolk, vanilla, and coconut extract (if utilizing), beating till mixed. With mixer on low velocity, step by step add remaining 1⅔ cups (208 grams) flour, beating simply till dough comes collectively and stopping to scrape paddle and backside and sides of bowl.
Swap to the dough hook attachment. With mixer on low velocity, add butter, one dice at a time, beating till mixed after every addition. Beat till dough is elastic and clean, 8 to 10 minutes.
Spray a big bowl with baking spray. Place dough in bowl, turning to grease prime. Cowl and refrigerate in a single day. (Alternatively, cowl and let rise in a heat, draft-free place [75°F/24°C] till doubled in measurement, 2 to three hours.)
Spray baking sheets with cooking spray. Lower 14 (5-inch) squares of parchment paper, and place on ready pans.
Prove dough onto a evenly floured floor, and roll to ½-inch thickness. Utilizing a 3×2½-inch oval or 3-inch spherical cutter dipped in flour, lower dough. Gently switch lower dough to ready parchment squares, spacing at the least 2 inches aside.
Reroll scraps; let dough stand for 10 minutes. Lower dough, and place on ready parchment squares. (Discard any remaining dough scraps.) Cowl with plastic wrap, and let rise in a heat, draft-free place (75°F/24°C) till puffed, about 1 hour (or about 1½ hours in case your dough was beforehand refrigerated).
In a 4- to 6-quart cast-iron Dutch oven or different massive heavy-bottomed saucepan, pour oil to a depth of two inches, and warmth over medium warmth till a deep-fry thermometer registers 350°F (180°C). Line a rimmed baking sheet with paper towels.
Utilizing parchment paper to choose up doughnuts, switch doughnuts in batches of two to three to sizzling oil, letting doughnuts gently fall off parchment into oil. (Don’t place parchment paper in oil.) Fry doughnuts in batches till golden brown, 1 to 2 minutes per aspect. Take away doughnuts from oil utilizing a spider strainer or slotted spoon, and let drain on ready pan. Dredge heat doughnuts in sugar to coat.
Spoon Haupia Filling right into a pastry bag fitted with a medium spherical piping tip (Ateco #803).
Utilizing a small paring knife, create a gap within the aspect of every doughnut; add Haupia Filling as desired. Greatest served similar day as made. Refrigerate in an hermetic container for as much as 2 days.
