In a mixing bowl combine 1/3 cup warm water (105–115°F), 2 1/4 teaspoons active dry yeast, and 1 teaspoon of the granulated sugar. Stir once and let sit 5 minutes, until the mixture is frothy.
Add the remaining granulated sugar (the rest of the 6 tablespoons after the 1 teaspoon used above), 2 slightly beaten large eggs, 3/4 cup warm milk, 1/2 cup melted butter, and 1/4 teaspoon salt to the yeast mixture. Stir to combine.
Gradually add 4 1/2 cups all-purpose flour to the wet mixture, stirring until a shaggy dough forms and the flour is incorporated.
Turn the dough onto a lightly floured surface and knead for several minutes (about 5–8 minutes) until the dough is smooth and elastic.
Shape the dough into a ball, place it in a clean bowl, cover with a clean towel, and let it rise in a warm place about 1 1/2 hours, or until doubled in size.
After the dough has doubled, gently deflate it. Roll the dough on a floured board to about 1/4 inch thickness. Cut out donut shapes with a donut cutter.
Place the cut donuts on a sheet of parchment paper, cover loosely with a towel, and let them rest and rise for an additional 30 minutes.
While the donuts are resting, heat canola oil in a heavy pan to 375°F. Use enough oil so the donuts can float while frying.
Fry the donuts in batches, 2–3 minutes per side, until golden brown. Turn once to brown both sides evenly. Remove donuts with a slotted spoon and drain on paper towels.
While still warm, optionally toss the donuts in granulated sugar or coat them with a donut glaze.