In a large mixing bowl, combine warm water, yeast, and 1 teaspoon of the sugar. Stir and let stand for about 5 minutes or until mixture is foamy.
Add the remaining sugar, eggs, milk, melted butter, and salt. Stir with a wooden spoon until well-combined.
Add the flour gradually while stirring, adding only enough until the dough has gathered in the center of the bowl. It should still be soft and not dry. You may not need the entire amount of flour.
Turn the dough over on a floured surface and knead until smooth, elastic, and has lost some of its stickiness. Use some of the remaining flour to prevent the dough from sticking too much on the board or on hands.
Shape the dough into a ball and place in a bowl. Cover the bowl with a kitchen towel and let the dough rise for about an hour, or until double in size.
Once risen, gently deflate the dough and divide it into 4 equal portions. Divide each portion into 4 smaller equal portions for a total of 16 pieces.
Using the palm of hands or a rolling pin, flatten each portion to form a rectangular dough about 6 inches in length and 4-inches in width.
Roll the flattened dough lengthwise to form a log. Using hands, roll the log back and forth against the kneading surface to elongate to 13 to 15 inches long.
Braid the log into a twisted rope by folding in the middle the log midways and overlapping each side over each other. Repeat with the remaining portions.
Arrange the braided dough in a single layer on a large baking pan at about an inch apart. Cover with a kitchen towel and let rise for 30 minutes or until puffy.
In a heavy bottom pan over medium heat, heat about 3 inches of cooking oil to 350 F.
Add the dough 2 to 3 at a time, starting with the puffiest ones, and deep-fry the dough for about 1 to 3 minutes or until golden. Adjust the heat from time to time to prevent the donuts from burning.
With a slotted spoon, remove fried donuts from the oil and drain on paper towels. In a shallow plate, toss the donuts in sugar to fully coat. Serve warm.