Sufganiyot (Jelly Doughnuts)

In a small saucepan, gently heat milk, butter, and sugar over medium-low heat until butter melts. Let cool about 10 minutes, until just warm to the touch. Pour into a large mixing bowl. Stir in yeast; let sit 5 to 10 minutes. Stir in eggs, vanilla, nutmeg, and salt. Stir in 2 cups flour until mixture is well incorporated, adding more flour until dough becomes smooth, soft, and pliable. Cover dough; let rise approximately 1-1/2 hours or until doubled in volume. Punch down dough with closed fist and knead gently in bowl. Let dough rest approximately 15 minutes. Using lightly floured hands, transfer dough to floured work surface and gently press down to 1/4-inch thickness. Using a cookie press or floured jar, cut out 3- to 4-inch circles. Collect scraps, re-form dough, and repeat cutting. Discard excess dough. Let dough circles rest on a floured, dusted surface about 30 minutes, or until doubled in size. Meanwhile, heat oil in a shallow-bottomed saucepan to approximately 350 degrees F. Gently slide 4 doughnuts at a time into oil, leaving space for expansion and even browning. Fry doughnuts until puffed and golden brown, about 1-1/2 to 2 minutes per side. Transfer to paper towels. Let cool 15 to 20 minutes. Pipe jelly or other filling into doughnuts with a pastry bag. Dust liberally with sugar.