Preheat the oven to 375F. Prepare the rice first. Measure out 1 cup of uncooked white rice, then cook it in a small sauce pot according to package instructions. The rice needs to be fully cooked before adding to the filling.
Next, prepare the mirepoix for the filling. Melt the 2 tablespoons butter in a large frying pan over medium heat. Add the grated carrots, diced onion and minced garlic. Saute the mixture, not browning, for 6 to 7 minutes, until the onion and carrot is tender. Transfer the mixture into a large mixing bowl.
Once the rice is done cooking, add it to the same mixing bowl. Follow with the seasonings, spices and herbs. Allow the rice to cool a bit, then add the egg and ground meat. Using your hands, mix all the filling ingredients together for a few minutes, until a uniform mixture forms.
Prepare the peppers: take a sharp paring knife and cut around the top of the pepper, removing the core and seeds. Tap the peppers a few times gently to get rid of any remaining seeds. I also like to remove the white membrane from the sides of the peppers; it can be bitter. Fill the peppers with the prepared filling, compacting the filling into the bottom of the peppers first. Arrange the stuffed peppers in a large cast iron pot, Dutch oven or copper braiser.
Next, prepare the braising sauce. Using the same frying pan from earlier, melt the remaining 1 tablespoon butter over medium heat. Add the diced onion and saute for 3 to 4 minutes until translucent. Deglaze the pan with a cup of beef broth, then add the tomato sauce and season the sauce with a generous amount of salt, black pepper and a bit of smoked paprika. Bring the sauce up to a simmer and cook for a few minutes. Add the remaining 3 cups beef broth and bring to a simmer again.
Pour the simmering sauce over the arranged stuffed peppers and sprinkle the top with more dill. The liquids will come up about a 1/3 to 1/2 way up the peppers; there is no need to add additional liquids. Cover the pan with a lid and place into preheated oven. Bake at 375F for 1 hour and 30 to 45 minutes; baking longer will make the peppers softer.
Allow the peppers to cool for about 30 minutes before serving; they will be extremely hot when they come out of the oven. Serve the peppers with a generous amount of sour cream on top and with a side of pickles and bread.