1. In a large bowl, stir together water, yeast, and pinch of sugar. Let stand 5 minutes, until foamy. Stir in remaining sugar, milk, 1/3 cup vegetable oil, eggs, salt, vanilla, and nutmeg. Add most of the flour to make a soft dough. Knead 5 to 8 minutes by hand or with a dough hook. Add the rest of the flour as needed to make a firmer dough that is smooth and elastic. Form into a ball and place in a large, lightly oiled bowl; rotate dough to coat with oil. Cover with plastic wrap and place in a warm spot until dough doubles in size, about 1 hour.
2. Punch dough down, cover, and let rise a second time, about 15 minutes. Grease two cookie sheets. Punch dough down again and divide in half. Transfer one half to a lightly floured countertop. Roll out 3/4-inch thick; cut out rounds with a 2 1/2-inch biscuit cutter. Place on a baking sheet two inches apart; cover with a floured kitchen towel. Repeat with remaining dough. Let dough rise until doubled in size, about 1 hour.
3. In a deep fryer or heavy Dutch oven, heat two inches of oil to a temperature of 380 degrees F (see Note). Lift dough circles with a metal spatula and gently flip them, top-side-down, into hot oil. Fry 2 minutes per side, until golden brown. Remove with a slotted spoon; drain on paper towels.
4. To fill, use a thin, sharp knife to pierce doughnuts from one side almost through to the other. Rotate knife to make a space inside doughnut. Place jam in a strong plastic bag (or a pastry bag fitted with a 1/4-inch tip), twist closed, and make a 1/4-inch cut in one corner. Pipe about 2 teaspoons of jam into each opening. Dust with sugar.