In a large bowl, combine warm milk, yeast, and 1 tablespoon of the sugar. Let sit until foamy, about 5 minutes. If it doesn't foam, your yeast is dead - start over with fresh yeast.

Add remaining sugar, melted butter, egg, flour, and salt. Mix with a dough hook on medium-low speed until a soft, sticky dough forms, about 4 minutes. It should pull away from the bowl but still cling to your finger. If too sticky, add flour 1 tablespoon at a time.

Transfer to a lightly oiled bowl, cover with plastic wrap, and let rise in a warm spot until doubled in size, 45 to 60 minutes. The dough should hold an indentation when poked.

Punch down dough and divide into 12 equal pieces. Roll each into a smooth ball and place in a greased muffin tin. Cover and let rise until puffy and touching, 20 to 25 minutes. Preheat oven to 375°F.

Bake until deep golden brown on top and an instant-read thermometer registers 190°F in the center, 13 to 15 minutes. Immediately remove from tin and let cool on a rack for 5 minutes.

Brush warm donuts all over with melted butter, then roll in sugar to coat. Fit a piping bag with a long narrow tip and fill with jam. Poke into the side of each donut, squeeze gently until you feel resistance, then pull out slowly. Serve warm or at room temperature.