Heat a dry medium saucepan over medium-low heat. Add 1/2 cup sweetened shredded coconut and cook, stirring constantly with a wooden spoon, until the coconut turns golden brown with some deep amber edges, about 4 to 5 minutes. Immediately transfer to a plate to cool. This goes fast at the end, so do not walk away.

Preheat oven to 350F. Whisk together flour, sugar, baking powder, and salt in a large bowl. In a separate bowl, whisk coconut milk, egg, melted butter, and vanilla until smooth. Pour wet ingredients into dry and fold until just combined - a few small lumps are fine. Fold in 1/4 cup of the toasted coconut (reserve the rest for topping).

Grease donut pans well with butter or nonstick spray. Transfer batter to a piping bag or zip-top bag with the corner snipped. Pipe batter into each cavity, filling about 3/4 full. Smooth tops with a damp fingertip if needed.

Bake until donuts spring back when lightly pressed and a toothpick comes out clean, 12 to 14 minutes. Cool in pans 5 minutes, then turn out onto wire racks to cool completely, about 15 minutes. Donuts must be fully cool before glazing or the glaze will slide right off.

Whisk powdered sugar, coconut milk, and vanilla in a medium bowl until smooth and pourable, like thick honey. Add more coconut milk by the teaspoon if too thick, or more sugar if too thin.

Dip the top of each cooled donut into the glaze, letting excess drip back into the bowl. Return to the wire rack and immediately sprinkle with the remaining toasted coconut. Let glaze set for 15 minutes before serving.