Place the boiling water, 1 tsp granulated sugar, dry active yeast, and milk in a mixing bowl.
Stir the mixture gently until combined.
Allow the yeast mixture to rest undisturbed for 10 minutes, or until it begins to activate.
In a separate mixing bowl, combine the melted butter, 2 3/4 cups all-purpose flour, and the remaining granulated sugar.
Stir the flour mixture until the ingredients are evenly incorporated.
Add the large egg and the activated yeast mixture to the flour mixture.
Mix until a soft dough begins to form.
Add additional flour gradually, as needed, until the dough pulls away from the spatula with ease.
The dough should remain slightly tacky to the touch at this stage.
Transfer the dough to a lightly floured surface.
Knead the dough for approximately 5 minutes, adding small amounts of flour as needed, until the dough is no longer tacky.
Place the kneaded dough into a greased bowl.
Cover the bowl and allow the dough to rise in a warm area for 1 hour, or until it has increased in size.
After the dough has risen, divide it into four equal sections.
Place each section on a lightly floured surface.
Roll each section into a log shape, ensuring each log is no more than 2 inches wide.
Cut each log into 15 equal pieces.
Roll each piece of dough into a ball.
Pour several inches of oil into a deep-sided skillet.
Heat the oil to 350°F.
Carefully place a portion of the dough balls into the hot oil, working in batches to avoid overcrowding.
Fry the doughnut holes for a few minutes, turning as needed, until they are evenly golden brown.
Remove the fried doughnut holes from the oil using a slotted spoon.
Transfer the doughnut holes to a paper towel-lined plate to drain.
Prepare the glaze by combining the hot water, powdered sugar, vanilla extract, and melted butter in a mixing bowl.
Whisk the glaze until smooth and fully combined.
Dip the warm doughnut holes into the glaze, coating them evenly.
Place the glazed doughnut holes on a plate or wire rack.
Allow the glaze to set for several minutes before serving.
Serve warm or at room temperature.