Yield: 8 doughnuts and 8 doughnut holes
Attach candy thermometer to a large heavy bottomed pot (cast-iron or Dutch oven, preferred). Fill pot about 3-inches deep with shortening or oil. Set over low-medium heat to 375 degrees. Prepare dough while oil heats.
In the bowl of a stand mixer fitted with a paddle attachment, combine 1 cup of flour with sugar, baking soda, baking powder, salt, and nutmeg. Add coconut milk, whisked eggs, and melted butter. Mix on low speed until smooth. Add remaining flour and continue to mix on low speed until thoroughly combined. Scrape down bowl as needed.
Transfer dough to a floured work surface. Dough will be rather sticky. Lightly flour the top of dough and pat or roll dough to a thickness of 3/4-inch. Using a floured 3 1/2-inch doughnut cutter (or round cookie cutters), stamp out doughnuts. Transfer doughnuts and doughnut holes to baking sheet. Gather scrape dough, knead together, and re-roll to stamp out more doughnuts.
Brush off any excess flour from the cut doughnuts. Carefully place about 3-4 doughnuts at a time into hot oil. Do not overcrowd the pot. Cook for 1 minute on each side, until golden in color. Cook doughnut holes for about 45 seconds on each side.
Drain cooked doughnuts on wire rack or paper towel lined baking sheet. Keep an eye on candy thermometer and adjust head accordingly to maintain a constant cooking temperature.
Mix together confectioners' sugar, coconut milk, and salt until mixture is smooth. Add additional coconut milk one teaspoon as a time if glaze is too thick to your liking.
Dip warm doughnuts into glaze, allowing excess glaze to drip. Immediately sprinkle shredded coconut over glaze. Set on a wire rack for glaze to set. Enjoy while doughnuts are warm, or enjoy at room temperature.