In a medium mixing bowl, combine the boiling water, 1 teaspoon of sugar, the active dry yeast, and the milk. Stir gently, then allow the mixture to rest undisturbed for 10 minutes, or until foamy.

In a separate large mixing bowl, combine the melted butter, 2¾ cups of the flour, and the remaining granulated sugar. Add the egg and the activated yeast mixture. Stir until the ingredients are fully incorporated and a sticky dough begins to form.

Gradually add additional flour, as needed, until the dough pulls away from the sides of the bowl and is soft but still slightly tacky. Transfer the dough to a floured surface and knead for approximately 5 minutes, adding small amounts of flour as necessary, until the dough becomes smooth and no longer sticky.

Lightly grease a large bowl with oil or nonstick spray. Place the kneaded dough in the bowl, cover with a clean kitchen towel or plastic wrap, and allow it to rise in a warm area for 1 hour, or until doubled in size.

After the dough has risen, divide it into four equal portions. Roll each portion into a log approximately 2 inches in diameter. Cut each log into 15 even pieces and roll each piece into a smooth ball to create doughnut holes.

Pour several inches of oil into a deep, heavy-bottomed skillet or pot and heat to 350°F (175°C). Working in batches, carefully lower the dough balls into the oil and fry for approximately 3 to 4 minutes, or until golden brown. Turn as needed to ensure even cooking.

Remove the doughnuts with a slotted spoon and place on a paper towel-lined plate to absorb excess oil. Allow to cool slightly before glazing.

In a bowl, whisk together the powdered sugar, hot water, vanilla extract, and melted butter until smooth and fully combined.

Dip the warm doughnut holes into the glaze and place on a wire rack or parchment paper to set for several minutes before serving.