Generously flour a clean work surface and lightly oil a medium-sized bowl.
In a large bowl or the bowl of a standing mixer, add the warm milk, then sprinkle the yeast over the milk. Using an electric mixer or the dough hook for a standing mixer, stir to dissolve and allow it to sit until it gets foamy, about 10 minutes. Add the butter, salt, and sugar, and stir or blend to combine. Blend in the egg and egg yolk, then beat in 2 cups of the flour. Beat until the dough starts to pull away from the sides of the bowl, then beat in another ½ cup of the remaining flour plus the nutmeg, mace, or cinnamon. When the batter has formed into a cohesive ball, turn it onto the floured work surface and knead the dough for two minutes. Add more flour as needed if the dough feels too sticky. Form the dough into a ball and place it in the oiled bowl. Cover with a dishtowel or plastic wrap and let sit in a warm, draft-free place until the dough has doubled in volume, about 1 hour.
Clean and re-flour your work surface. Turn the dough onto the floured surface and roll it out with a floured rolling pin to a ½-inch thickness. With a 3-inch biscuit or cookie cutter, cut as many rounds as you can, minimizing the space between each round. Gather the scraps, let them sit for another 15 minutes, then roll the dough out again and cut as many more circles as possible. Place the circles on a very lightly floured baking sheet with a couple of inches between each doughnut, cover with a clean dishtowel or loose plastic wrap, and let sit for about 1 hour, until they are quite puffy.
About 15 minutes before you are going to fry the doughnuts, place paper towels on a clean surface. Make the sugar coating: in a shallow bowl, mix together the 1 cup sugar with the cinnamon. And pour at least 2 inches of oil into a deep skillet or pan. Heat to 375 F over medium heat.
Use a spatula to transfer two or three doughnuts into the pan. They will rise and bob on the surface; fry for about 1 minute, until golden brown on the underside, then flip them and cook until the second side is golden brown, another 1 to 2 minutes. You can occasionally gently press the doughnuts down into the oil to cook the sides evenly. Remove the doughnuts, allowing excess oil to drain back into the pan, and let them rest for a minute on the paper towels.
Use a chopstick or wooden dowel to poke a hole into the side of each doughnut, and as you slide it in, give it a wiggle to create a small pocket in the center of the doughnut. Fill a pastry bag or study plastic bag with the jelly. If you are using a plastic bag, cut a very small triangle from one of the bottom corners of the bag. Insert the pastry bag or snipped corner of the plastic bag into a hole on the side of the doughnut. Gently squeeze a couple of teaspoons of jelly into the middle of the doughnut. Remove the bag carefully from the doughnut, and repeat until all of the doughnuts are filled.
Serve warm or at room temperature.
Notes
You can use any type of jam you like, and this is a great way to play around with those jams or jellies that you may have picked up at a farmers market or specialty store or on your travels. Strawberry and raspberry jelly are very popular filling flavors, but you can really use anything from apricot to mixed berry to pear, whatever you fancy.