In the bowl of a stand mixer fitted with the dough hook, stir the 1/4 cup sugar into the 1 1/8 cups warm coconut milk (or warm regular milk if using). Sprinkle the 2 1/4 teaspoons yeast over the milk-sugar mixture, stir gently, and let sit 10 minutes. (If using active dry yeast, wait until the mixture is foamy; if using instant yeast it may not foam but proceed after 10 minutes.)
While the yeast proofing rests, melt the 1 1/4 stick unsalted butter. Let the melted butter cool briefly so it is warm but not hot, then whisk it into the 2 beaten large eggs a little at a time to avoid cooking the eggs.
Add the egg-and-butter mixture to the yeast mixture. Turn the mixer to medium-low and mix a couple of minutes until combined.
With the mixer running on medium-low, add the 1/4 teaspoon salt and the 4 cups all-purpose flour in ~1/2-cup increments, scraping the bowl as needed, until all the flour is incorporated.
Stop the mixer, scrape the bowl, then run the mixer again on the same speed for 5 minutes to develop the dough. The dough should be smooth and slightly tacky but not sticky; if needed, scrape and check the texture.
Transfer the dough to a lightly oiled bowl, turn the dough to coat it in oil, cover the bowl tightly with plastic wrap, and refrigerate for at least 8 hours or overnight.
Remove the chilled dough from the refrigerator and turn it out onto a lightly floured surface. Roll the dough to 1/4 to 1/3 inch thickness.
Using a 3-inch round cutter, cut as many rounds as possible. Re-roll scraps and cut additional rounds. Use a 1-inch cutter to cut holes from the centers to form doughnuts. Place finished doughnuts and holes on a floured baking sheet, spacing them slightly apart.
Cover the doughnuts with a large towel and set them in a warm, draft-free spot to rise until puffy and airy, about 50–60 minutes.
While the doughnuts are proofing, make the Espresso Glaze: in a medium bowl whisk together 1/4 cup coconut water, 3/4–1 cup powdered sugar, and 2 tablespoons strong black coffee until smooth. Set the glaze aside.
When the doughnuts have risen, heat a few inches of neutral frying oil in a large saucepan over medium-high heat until the oil reaches 350°F (use a thermometer). Fry 2–3 doughnuts at a time, turning once, until golden brown and cooked through — about 2–3 minutes per side (about 4–6 minutes total). Adjust heat to maintain 350°F so they brown without burning.
Transfer fried doughnuts to paper towels to drain briefly. While still warm, dunk each doughnut thoroughly in the Espresso Glaze, then sprinkle each with 2 tablespoons unsweetened shredded coconut. Place glazed doughnuts on a wire rack to let excess glaze drip off. Store in an airtight container if not grilling immediately (best within one day).
Preheat the grill to medium heat while you make the ganache.
To make the Mocha Coconut Ganache, place the 5 ounces milk chocolate and 7 tablespoons coconut milk in a small saucepan over low heat. Stir constantly until the chocolate has melted and the mixture is smooth. Remove from heat, whisk in 2 teaspoons instant espresso powder until dissolved, then stir in 1 teaspoon vanilla. (Alternatively, melt the chocolate and coconut milk in the microwave in 30-second intervals, stirring between intervals, then add espresso and vanilla.)
Lightly coat both sides of each glazed doughnut with cooking spray or brush very lightly with a neutral oil (to prevent sticking). Place doughnuts on the preheated grill and grill, covered if possible, until lightly browned, about 30 seconds to 1 minute per side. Watch closely to avoid burning.
Serve the grilled doughnuts warm with the Mocha Coconut Ganache for dipping.