Warm the canned coconut milk to about 110°F (warm to the touch but not hot). In a large bowl, combine the warm coconut milk, sugar, and yeast. Stir gently and let it sit for 5-10 minutes until it becomes foamy, indicating the yeast is activated.
Add the beaten eggs and melted unsalted butter to the yeast mixture. Stir to combine. In a separate bowl, whisk together the flour and salt, then gradually add the dry ingredients to the wet mixture. Use a stand mixer with a dough hook or your hands to knead the dough for about 8-10 minutes until smooth and elastic. If the dough feels too sticky, add a little more flour, one tablespoon at a time.
Lightly grease a clean bowl and place the dough inside. Cover with a damp cloth or plastic wrap and let it rise in a warm place for about 1 to 1.5 hours, or until doubled in size.
Once risen, punch down the dough to release air. On a floured surface, roll the dough to about 1/2-inch thickness. Use a doughnut cutter or two round cutters to cut out doughnut shapes. Transfer the doughnuts to a parchment-lined tray, cover, and let them rise for another 30-45 minutes.
Preheat your grill or grill pan to medium heat. Lightly oil the grates or pan surface to prevent sticking.
Place the doughnuts on the grill and cook for 2-3 minutes on each side. Watch carefully to avoid burning; you want nice grill marks and a crisp exterior while keeping the inside soft and fluffy. Remove and let cool slightly on a wire rack.
In a small bowl, whisk together powdered sugar, strong black coffee, and vanilla extract until smooth. Stir in the unsweetened shredded coconut. The glaze should be thick but pourable; adjust consistency with a bit more powdered sugar or coffee as necessary.
In a saucepan over low heat, combine milk chocolate, coconut milk, and instant espresso powder. Stir continuously until the chocolate melts completely and the mixture is smooth and glossy. Remove from heat and stir in vanilla extract.
Dip the grilled doughnuts into the espresso glaze, allowing excess to drip off. Place them back on the cooling rack. Once the glaze sets slightly, drizzle or spread the mocha coconut ganache over the top. You can layer as much ganache as you like for a richer taste.
Let the ganache set for a few minutes before serving. These doughnuts are best enjoyed fresh with a warm cup of coffee or your favorite beverage.