In large mixing bowl, stir together 1/2 cup warm water, yeast and pinch sugar. Let stand for several minutes. Stir in remaining 1/3 cup sugar, milk, vanilla, eggs, oil, salt and most of the flour to make a soft dough. Knead for 5 to 8 minutes by hand or with a dough hook, adding more flour as needed to form a firmer dough that is smooth and elastic. Place dough in greased bowl. Place bowl in plastic bag and seal. (If making dough a day in advance, refrigerate at this point.) Let rise for about 1 hour. Gently deflate. (If dough was refrigerated, let come to room temperature for about 40 minutes before proceeding). Pinch off pieces of dough and form into small balls, a little larger than a golf ball. Alternately, roll dough about 3/4 inch thick and cut into 2 1/2- or 3-inch rounds. Cover doughnuts with a clean tea towel; let sit for 20 to 30 minutes. Heat about 4 inches oil in deep fryer or heavy Dutch oven to about 385 degrees. Fry a test doughnut; oil should bubble. Fry until underside is deep brown. Turn once; fry until other side is deep brown. Total frying time should be 1 1/2 to 3 minutes. Remove with slotted spoon; drain on paper towels. Check middle of doughnut to make sure inside is cooked. If not, reduce heat of oil slightly. Fry remaining doughnuts, 3 or 4 at a time. To fill, make a small opening in each doughnut; spoon in jam or jelly. (Instead of filling doughnuts, you can shake them in a paper bag with granulated or powdered sugar.)