Put the milk in a large bowl—use your stand mixer if you have a dough hook—and sprinkle with the yeast. Let it sit for a few minutes, until it gets foamy. Add about 3 cups of the flour along with the sugar, eggs, butter and salt and stir until you have a sticky dough. Add a bit more flour and knead until the dough is smooth and elastic, but still very tacky—the flour will absorb some of the liquid and smooth out as it rests. Cover and let rise for about an hour and a half, or until it doubles in size.
Roll or pat the dough out about 1/2-inch thick and cut into about 2 inch rounds with a cookie cutter or the rim of a glass. (Re-roll the scraps if you like—I like to cook them as is for wonky sugary doughnut bits that always wind up looking kind of like animals.) Cover the rounds with a tea towel and let rest for about half an hour.
When you’re ready to cook, set a heavy, shallow pot or Dutch oven over medium-high heat and add about two inches of oil; heat it until it’s hot, but not smoking—a scrap of bread or dough dipped in should sizzle, and if you have a thermometer it should read about 350F. Cook the doughnuts in batches, without crowding the pot—turn them with a slotted spoon or tongs for 2-3 minutes, or until they turn deep golden. Transfer to a paper towel-lined plate, and when they’re cool enough to handle (but still quite warm), roll them around in a shallow dish of sugar to coat.
Put the jam into a piping bag or zip-lock bag and snip off one corner. Poke a knife into the side of each doughnut, stick the end of the bag into the hole and gently squeeze in some jam - you'll feel it getting heavier. (Alternatively, slice each doughnut halfway open and spoon some in.) Serve warm, if at all possible.