In a small bowl, combine the warm water and 1 teaspoon of granulated sugar. Sprinkle the active dry yeast on top and let it sit for about 5-10 minutes, or until it becomes frothy.
In a large mixing bowl, combine the warm milk, melted butter, slightly beaten eggs, and the remaining 5 tablespoons of granulated sugar. Mix well until everything is combined. Next, add the frothy yeast mixture and stir until fully incorporated.
Gradually sift in the all-purpose flour and salt. Stir the mixture until a dough begins to form. Knead the dough until it's smooth and elastic.
Cover the bowl with a clean kitchen towel and let the dough rise in a warm place for about 1-1.5 hours, or until it has doubled in size.
Once the dough has risen, punch it down, turn it out onto a floured surface, and roll it out to about 1/2 inch thick. Use a cookie cutter to cut out donut shapes.
In a deep frying pan, heat about 2-3 inches of canola oil to 350°F. Use a thermometer to monitor the temperature.
Carefully place a few donuts in the hot oil, frying for about 1-2 minutes on each side, or until golden brown. Use a slotted spoon to remove the donuts and place them on a cooling rack.
Once the donuts have cooled slightly, roll them in granulated sugar or dip them in your favorite glaze. Allow the glaze to set before serving.