In a small bowl, combine the bread crumbs and the milk; stir with a fork and let stand for 5 minutes. Put 3 tablespoons of the butter in a small skillet and let it melt over medium heat. Add the shallots, garlic, and caraway; season with salt and pepper. Saute until softened but not browned, about 2 minutes. Put the ground beef and pork in a large bowl. Add the sweated shallot mixture, egg, parsley, and dill; season with salt and pepper. Squeeze the milk out of the soaked bread crumbs, add the bread to the meat mixture, and mix well using your hands. Pinch off about 1/2 cup worth of the meat mixture and roll it around in your moistened hands to shape it into a cue ball-sized meatball. Continue forming the meatballs and putting them on a platter; there should be about 20 meatballs.
Melt 2 tablespoons of the butter in a large skillet over moderate heat. When the foam starts to subside, add the meatballs in batches so you don't overcrowd the pan. Saute the meatballs until well-browned on all sides, about 7 minutes total. Remove the meatballs to a platter lined with paper towels as they are done.
Discard most of the fat from the skillet and return it to the heat. Add the remaining tablespoon of butter and swirl it around to coat the pan. Sprinkle in the flour, and stir with a wooden spoon to dissolve into the fat. Pour in the chicken broth, and stir with a wooden spoon to loosen the bits from the bottom of the pan. Simmer and stir until the liquid is reduced and a sauce starts to form; season with salt and pepper. Lower the heat and stir in the sour cream. Return the meatballs to the sauce and stir in 1/4 cup of the lingonberry jam until combined. Simmer until the sauce thickens slightly and the meatballs are heated through, about 15 to 20 minutes. Shower with chopped parsley and transfer to a serving bowl. Serve the Swedish meatballs with additional lingonberry jam on the side.