Blanch the cabbage leaves: Bring a large saucepan of salted water to a boil. Add whole leaves (or carefully peel layers). Cook 2–5 minutes until softened and flexible. Then transfer to a warm plate, pat dry, and cover to keep pliable. Reserve the core and any torn bits for chopping.

Sauté the aromatics: Melt butter in a skillet over medium heat. Add onion, carrot, and garlic; sauté until tender and fragrant, 4–6 minutes. Salt to taste. Next, divide the mixture: place two-thirds in a bowl for the filling, and reserve one-third for the baking dish.

Mix the filling: In a large bowl, combine ground chicken, cooked rice, the reserved sautéed onion/carrot/garlic (two-thirds), beaten egg, dill, salt, and pepper. Mix gently until just combined. Do not overwork; the rolls will stay tender.

Assemble the cabbage rolls: Trim thick ribs from leaves if needed. Place a mound of filling near the base of each leaf. Then tuck in sides and roll snugly to enclose the filling. Keep rolls seam-side down. Work until all filling is used.

Prepare the baking dish: Dice leftover cabbage leaves (and any trimmings). Scatter them into the bottom of a 9×13-inch baking dish. Then spoon the reserved one-third onion/carrot/garlic over the chopped cabbage. Arrange rolls on top, seam-side down, in tight rows.

Make the sauce and bake: Whisk chicken broth, tomato paste, and dill. Pour over the rolls. Add bay leaves. Next, add enough water to just cover the rolls, and season the liquid with salt and pepper to taste. Cover tightly with foil. Bake at 350°F for 1½ to 2 hours, depending on how soft you like them. Finally, remove from the oven, rest 10 minutes, and serve warm with sour cream if you like.