1. In a small bowl, mix together the warm milk and yeast. Let sit for 5 minutes until frothy.
2. In a stand mixer fitted with a dough hook, mix together the butter, sugar, flour, salt and egg. Next, add in the activated yeast and milk. Put the machine on medium and beat with the dough hook for 5 minutes until a nice dough has formed. Lastly, add the basil, oregano, sun-dried tomatoes, and mozzarella. Mix until incorporated.
3. Cover the bowl with a clean kitchen towel and let rise in a warm place for 1–2 hours.
4. Once risen, remove the dough from the bowl and roll into a 10-inch rectangle. Cut out 2-inch circles. Re-roll the dough, keeping the same 1 ½-inch thickness until all the dough is used up and you’ve created about 15 rounds.
5. Place one cube of cheese in the center of each round and pinch to close the dough over the the cheese, being sure to completely seal the cheese inside. Roll the dough back into a round disc shape and continue with remaining dough. Once all the rounds have been filled, place on parchment paper and cover with a clean kitchen towel for an additional 20 minutes until puffed up again.
6. Meanwhile, heat your oil. Place oil in a medium pot and attach a candy thermometer to the side. Heat oil until it reaches 325ºF.
7. Put 3–4 doughnuts in the oil at a time, frying for about 3 minutes on the first side, and 2–3 minutes on the other. Keep checking to ensure your oil remains at 325ºF and that your doughnuts are deeply golden brown.
8. Remove doughnuts with a slotted spoon to a plate lined with paper towels. Continue with remaining rounds
9. Serve fresh from the oil as they are best hot!