Preheat oven to 400°F (200°C). Butter and flour a 6-well doughnut pan*.

In the bowl of a stand mixer fitted with the paddle attachment, beat sugars and egg at medium speed until lightened in color and well combined, 1 to 2 minutes, stopping to scrape sides of bowl. Add melted butter, oil, and extracts; beat at medium speed until well combined, about 1 minute.

In a small bowl, whisk together flour, baking powder, and salt. In another small bowl, whisk together coconut milk and 3 tablespoons (45 grams) room temperature water. With mixer on medium-low speed, add flour mixture to sugar mixture alternately with coconut milk mixture, beginning and ending with flour mixture, beating just until smooth and combined after each addition and stopping to scrape sides of bowl.

Spoon half of batter (about 1 cup [277 grams]) into a large pastry bag; cut a ½-inch opening in tip. Pipe batter evenly into prepared wells, refilling bag as necessary. (Use a paper clip or binder clip to hold pastry bag tip shut when refilling with batter.) Firmly tap pan on a kitchen towel-lined counter several times to settle batter and release any air bubbles.

Bake until a wooden pick inserted near center comes out clean, 8 to 10 minutes. Let cool in pan for 5 minutes. Invert doughnuts onto a wire rack placed over a parchment paper-lined baking sheet, and let cool completely.

Dip 1 doughnut halfway into Coconut Glaze; lift straight out, and swirl in a circular motion, letting excess drip off. Quickly turn and place, glaze side up, on wire rack. Repeat with remaining doughnuts. Garnish with coconut, if desired. Serve immediately.