Dissolve the yeast in the warm water; warm means about 98°F. Let this sit 10 minutes, then add it to a large bowl with the milk, salt, beaten egg, and sugar. Mix well.
Add 2 cups of flour and mix until well combined. Pour in the melted butter and incorporate that into the dough. Now stir in the remaining flour until you have a soft but workable dough; you might need an additional 1/2 cup of flour. Knead it until smooth, about 5 minutes.
Grease a bowl, then roll the dough ball in the oil, then cover the bowl in plastic wrap. Let this sit in a warm place for 1 hour. Punch the dough down, reform into a ball and let it sit, covered, for another 30 minutes. Separate into 12 equal sized balls and set them in the bowl, covered.
FILLING
When the dough begins its first resting period, cook the onion and meat in butter until browned, adding the salt and white pepper near the end of cooking. Move the meat and onions to a bowl to cool. Mix in the cooked rice and the chopped boiled egg and caraway seeds if using.
TO FINISH
Add about 1 to 2 inches of canola, vegetable or other high smoke point oil to a large frying pan set over high heat. You want the oil to hit 350°F.
As the oil is heating, either roll out a disk with a rolling pin or use a tortilla press to flatten out a ball of dough. Add a couple tablespoons of the filling to half of the disk, then fold over the other half to form a half-moon shape. Press to seal with your fingers, then with the tines of a fork. Set each pie aside until the oil is hot.
Turn the oven to 200°F and set a baking sheet in it. Set a cooling rack over the baking sheet.
I find I can fry two pies at a time while making the rest, but I make a lot of pies. Start slowly if you're new at it. You only need to fry the pies until they are pretty and golden, maybe 2 to 3 minutes per side. Flip only once. Set the finished pies on the rack in the oven while you do the rest.
Serve hot out of the fryer, or save them for lunch the next day!
Notes
Make sure the rice is short or medium grain! Look for Japanese rice, or risotto rice like arborio. Those are easy to find in supermarkets.