1. In a small bowl, combine yeast, warm water, and 1 teaspoon granulated sugar. Let stand until foamy, about 10 minutes.

2. Place flour in a large bowl. Form a well in the center; add eggs, yeast mixture, remaining 1/4 cup granulated sugar, the butter, nutmeg, and salt. Using a wooden spoon, stir until a sticky dough forms. Turn out dough onto 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, and let stand in a warm, draft-free spot until doubled in size, 1 to 1 1/2 hours.

3. Turn out dough onto a lightly floured work surface, and, using a lightly floured rolling pin, roll out dough to 1/4-inch thickness. Using a 2 1/2-inch round cutter, cut out 20 rounds, re-rolling scraps as necessary. Transfer to a lightly floured baking sheet, cover with plastic wrap, and let rise 20 minutes.

4. Whisk together confectioners' sugar and milk in a medium, shallow bowl until smooth. Cover glaze with plastic wrap and set aside until ready to fill doughnuts.

5. In a medium saucepan over medium heat, heat oil until a deep-fry thermometer registers 370 degrees. Using a slotted spoon, carefully slip 4 dough rounds into the oil. Fry until golden, turning once, about 1 minute on each side. Transfer to a paper towel-lined baking sheet to drain. Repeat process with remaining dough rounds.

6. Fill a pastry bag fitted with a coupler and small bismark pastry tip for filling (such as #230) with the jam. Using a chopstick or a wooden skewer, make a hole in the side of each doughnut. Fit the pastry tip into the hole, and pipe about 2 teaspoons jam into doughnut. Repeat with remaining doughnuts and jam.

7. Dip tops of doughnuts in the glaze to cover. Let stand, glaze-side up, until set.