Melt the milk and the butter together in a small bowl. Set aside to cool slightly (110°F).
2. In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, sugar, yeast and salt. Add the milk mixture, the eggs and egg yolks and mix on low just until combined. Switch to the dough hook and knead the dough on low speed until smooth and elastic, about 6 minutes. The dough should be workable but still somewhat sticky. Add a few more tablespoons of flour if necessary but resist the urge to add too much.
Transfer the dough to a buttered bowl, cover with plasitc wrap, and set it in a warm place to double, 60 to 90 minutes.
Tip the dough out onto a lightly floured surface and knead it once or twice to expel the air. Roll the dough into a 91/2 by 12 1/2 inch rectangle. It should be about 1/2 inch thick. Using a 3 inch round cookie cutter, cut out 12 dough rounds and set them on two floured rimmed baking sheets. Lightly cover the sheets with plastic wrap and set in a warm place until almost doubled, 60 to 90 minutes.
Line a rimmed baking sheet with a few layers of paper towels. Add the oil to a medium, heavy-bottomed pot. Attach a candy thermometer to the side of the pot and heat the oil to 350°F. Carefully add 2 to 3 doughnuts to the oil and fry them until golden brown, 2 to 3 minutes per side. Using a slotted spoon, transfer the doughnuts to the paper towels. Repeat with the remaining dough.
Toss the warm doughnuts in the powdered sugar. Use a chopstick to poke a hole into one side of each doughnut. Twist the chopstick around to make some room in the center of the doughnut. Be careful not to poke through the other side. Transfer the jam to a pastry bag fitted with a small round tip. For each doughnut, insert the tip of the pastry bag into the hole and gently squeeze to fill.
Alternatively, instead of powdered sugar, make a glaze by whisking the powdered sugar together with the cream and some food coloring. Top each filled doughnut with some glaze and top with sprinkles. Let the glazed doughnuts stand at room temperature for a few minutes to set.