In a small bowl, combine warm milk and butter, stirring to melt. Sprinkle yeast over the milk and stir to combine. Let sit until yeast softens and begins to foam, about 10 minutes.
In the bowl of a large food processor fitted with a dough blade, combine flour, sugar, and salt. Add milk and egg yolks. Process until a smooth dough forms.
Place the dough in a lightly greased bowl, turning to coat. Cover and let rise until doubled, about 1 hour. Time permitting, place bowl in refrigerator after doubling to chill, making the dough easier to roll out.
On a lightly floured surface, roll the dough until it is 1/2 inches thick. Cut with a 3 inch circular cutter and allow the cut doughnuts to rise until puffy, about 30 minutes.
In a large saucepan, heat 3 inches of canola oil over medium heat to maintain a temperature of 375 degrees Fahrenheit. Fry the doughnuts, about 3 at a time to avoid overcrowding and top side down first, until they are golden brown on both sides, about 4 minutes per side. Remove with a large slotted spoon to a towel lined plate.
Fit a pastry bag with a small to medium tip and fill with jam. Use a toothpick to poke a hole in the side of each doughnut and pipe in the jam. Top with powdered sugar.