Preheat the oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking dish.
Melt butter in a skillet over medium-high heat. Add onion and cook until tender, about 5 minutes. Remove from the heat and allow to cool.
While the onion is cooling, place bread crumbs into a large bowl. Pour milk over bread crumbs and let stand until all liquid has been absorbed, about 5 minutes. Stir in cooked onion, ground beef, ground pork, egg, salt, and pepper until well combined.
Cover the bottom of the prepared baking dish with two large beet leaves; set aside two more leaves for the top. Place the remaining beet leaves onto a flat work surface; add a heaping teaspoon of the meat mixture to the wide end (bottom) of each leaf. Working one at a time, roll the leaf up around the filling and place into the baking dish. Pour chicken broth over the rolls, then place the last two reserved leaves over the top.
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. Drain liquid from the bottom of the baking dish into a small saucepan. Whisk flour and cream together in a bowl until smooth; stir 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.
Place the filled rolls into a serving dish and pour the sauce over top. Sprinkle with chopped dill.