Cut the core from the cabbage. Bring a very large amount of water to a boil in a deep stockpot and carefully drop the whole head of cabbage into the boiling water. Parboil for 10 to 15 minutes, until leaves are soft and pliable. Remove head of cabbage from water with a two-pronged fork inserted into the coring hole, with a wide spatula for supporting the bottom. Drain and let cool.
When cabbage has cooled, carefully separate the leaves and stack them together. With a small, sharp knife cut and remove an inverted "V" out from the thick part of the ribs, so the leaves will lie flat.
Melt the butter in a nonstick pan and saute the onions until wilted Add the mushrooms and barley and cook for 3 minutes. Add lemon juice through parsley and cook for 1 minute, then set aside to cool.
Spoon about 1 tablespoon of the filling onto the center of each cabbage leaf; smaller leaves take less, larger leaves take more. Fold the sides of each leaf over the filling first, then roll up from the cut stem side to enclose the filling, like making a loose burrito. Do not over-fill, or roll too tightly, or they will burst from the expansion while cooking!
Place the rolls seam-side-down in one layer in a large skillet or saute pan with deep sides. Slowly pour the hot stock around the rolls, cover, and simmer over very low heat for 1 hour and 15 minutes.
Remove the rolls and place them in one layer on a serving platter and keep them warm while preparing the sauce.
Pour the remaining cooking liquid into a cup; there should be about 1/3 cup. Strain the liquid, wipe out the skillet, and return the strained liquid to it. Stir in the sour cream and the dill. Cook over very low heat, stirring, for 2 minutes. Spoon the sauce over the cabbage rolls and serve at once.