Mix the yeast with 2 tablespoons of the milk and 2 tablespoons sugar and let sit until foamy, about 5 to 10 minutes then add to the flour with the remaining milk.
In a mixing bowl combine 4 cups flour and salt then add the yeast.
Add the eggs and butter to the flour mixture.
Begin to mix the ingredients into a soft, but not sticky, dough. Add a little extra milk if needed, 1 tablespoon at a time or work in extra flour if needed, 1 tablespoon at a time.
Turn the dough out onto a lightly floured surface and knead until smooth and elastic.
Transfer the dough to a lightly oiled bowl, cover, and let rise until doubled in size, about 2 hours.
Punch down the risen dough then turn out onto a lightly floured surface and knead for a few seconds.
Lightly flour a rolling pin then gradually roll out the dough to about 1/2" thick. Cut out into 3 to 4-inch rounds with a lightly floured cookie cutter. Re-roll the scraps to make more rounds.
Place the doughnuts on lightly floured baking sheets, spacing them apart so they don't touch when rising, and cover lightly with a dry towel. Let rise in a warm spot until doubled in size, about 20 minutes.
Fill a fryer 3-inches high with vegetable oil and heat at about 350 F. Fry the doughnuts, a few at a time, until golden and puffed, turning frequently, about 5 minutes.
Lift the doughnuts from the oil and place on a plate lined with a paper towel.
Let cool slightly and fill a pastry bag, fitted with a 1/2" round tip, with jam.