Preheat oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking dish.
Melt the butter in a skillet over medium-high heat, and cook the onion until tender, about 5 minutes. Remove the skillet from the heat; allow to cool. Place the bread crumbs into a large bowl; pour the milk over the crumbs. Let stand until the crumbs have absorbed the liquid, about 5 minutes. Stir in the cooked onion, ground beef, ground pork, egg, salt, and pepper until well combined.
Cover the bottom of the prepared baking dish with 2 large beet leaves; set aside 2 more leaves for the top. Lay remaining beet leaves for filling onto a flat work surface, and place a heaping teaspoon of the meat mixture at the wide end (bottom) of each leaf. Roll the leaf up, enclosing the filling, then place into the baking dish. Repeat with the remaining leaves and filling mixture. Pour the chicken broth over the filled rolls, then place the last 2 reserved leaves over the top of the rolls.
Bake in the preheated oven until the beet leaves are tender and the filling is cooked through, about 1 hour. An instant-read thermometer inserted into the center should read at least 160 degrees F (72 degrees C).
Remove the dish from the oven, and pour the liquid from the bottom of the baking dish into a small saucepan. Whisk the flour and cream together in a bowl until smooth, and stir the cream mixture gradually into the pan juices. Bring to a bare simmer over medium-low heat, and cook gently until thickened, about 5 minutes. Do not boil. To serve, arrange the filled rolls in a serving dish and pour the sauce over the rolls. Sprinkle with chopped dill.