Whisk the milk, yeast, and 1 teaspoon of sugar together in a small bowl or liquid measuring cup. Set it aside for about 10 minutes, until frothy.
In a large mixing bowl, beat the butter and remaining 1/4 cup of sugar together with an electric mixer at high-speed until creamy, about 1 minute.
Beat in the vanilla and salt.
Begin adding the yeast mixture to the butter mixture, beating in a bit at a time at low speed. Be careful to avoid splashing.
Begin adding flour, about 1/2 cup at a time, beating in each addition. Switch to a spoon when the dough becomes too thick for the mixer to handle. Continue adding flour until a soft dough forms (about 3 1/2 cups total).
Transfer the dough to a lightly floured work surface, and knead until smooth and elastic, about 5 minutes.
Transfer the dough to a lightly oiled mixing bowl and cover it with a damp tea towel. Place the bowl in a warm spot to rise until doubled in size, about 1 hour.
Punch the dough down, then transfer it to a lightly floured work surface. Roll the dough to about 1/2 inch thick.
Use a doughnut cutter or biscuit cutter to cut the dough into doughnut shapes.
Reroll the excess dough and repeat until all of the dough has been cut.
Transfer the doughnuts to a parchment paper-lined baking sheet. Cover them with a damp tea towel and place the baking sheet in a warm spot to rise until puffy, about 1 hour.
Fill a large pot or Dutch oven with 1 1/2 to 2 inches of oil and place it over medium-high heat.
Heat the oil to about 350°F. Use a frying thermometer to monitor the temperature.
Add a few doughnuts to the oil, being very careful not to splash. Avoid crowding the pot.
Let the doughnuts cook for 1 to 2 minutes, flip and cook 1 to 2 minutes more, until golden brown on both sides and puffy.
Use a slotted spoon or tongs to remove the doughnuts from the pot. Transfer them to paper towel-lined plates to drain and cool.
Repeat until all of the doughnuts are cooked.
Whisk all ingredients together in a small bowl. Add more powdered sugar if it seems too thin, more milk if it's too thick. You want it to be somewhat runny, but thick enough to coat the doughnuts.
Melt the chocolate chips by microwaving in 30 second increments, stirring between each zap.
Stir in the butter, until melted, followed by the maple syrup.
When the doughnuts have almost completely cooled, dip each one in the glaze, to coat both sides.
Transfer the doughnuts to a wire rack for the glaze to set. I recommend placing a plate or paper towel under the rack to catch any drips.
If you're frosting your doughnuts, wait for the glaze to fully set, then dip the tops in the frosting. Optionally top your doughnuts with sprinkles, nuts, or coconut.
Allow the frosting to set, then serve.
Adapted from Delish.