In a small bowl, combine yeast, warm water, and 1 teaspoon sugar. Set aside until foamy, about 10 minutes.
Place flour in a large bowl. Make a well in the center; add eggs, yeast mixture, 1/4 cup sugar, butter, nutmeg, and salt. Using a wooden spoon, stir until a sticky dough forms. On a well-floured work surface, knead until dough is smooth, soft, and bounces back when poked with a finger, about 8 minutes (add more flour if necessary). Place in an oiled bowl; cover with plastic wrap. Set in a warm place to rise until doubled, 1 to 1 1/2 hours.
On a lightly floured work surface, roll dough to 1/4-inch thickness. Using a 2 1/2-inch-round cutter or drinking glass, cut 20 rounds. Cover with plastic wrap; let rise 15 minutes.
In medium saucepan over medium heat, heat oil until a deep-frying thermometer registers 370 degrees. Using a slotted spoon, carefully slip 4 rounds into oil. Fry until golden, about 40 seconds. Turn doughnuts over; fry until golden on other side, another 40 seconds. Using a slotted spoon, transfer to a paper-towel-lined baking sheet. Roll in sugar while warm. Fry all dough, and roll in sugar.
Fill a pastry bag fitted with a #4 tip with jam. Using a wooden skewer or toothpick, make a hole in the side of each doughnut. Fit the pastry tip into a hole, pipe about 2 teaspoons jam into doughnut. Repeat with remaining doughnuts.