To begin the lobster filling: Melt the butter in a saute pan which is just large enough to accommodate the lobster tails. The butter should be very hot, but not sizzling (between 165 and 180 degrees F). Add the tails to the pan, meat side down - shell side up, and poach until the lobster is opaque. Remove to a utility platter and let cool. (Pour the melted butter into a bowl for use in other elements in this recipe.) Return 2 tablespoons of the reserved melted butter to the same saute pan and saute shallots and shiitake mushrooms until tender. Add crabmeat and scallions and heat for a few minutes, trying to avoid breaking up lumps. Set aside. Return to the lobster tails, loosen them from the shell. Reserve the shells for use in the sauce. Slice the tail meat lengthwise down the middle and into 4 slices crosswise.
For the wrappers: Steam the cabbage leaves until tender but not mushy. Remember, the leaves will still have to be firm enough support the lobster filling.
To begin the sauce: Combine the milk, onion, bay leaf, cloves, nutmeg, and lobster shells in a heavy-gauge, non-corrosive saucepan over medium heat. Spoon 2 tablespoons of the reserved melted butter (from poaching the lobster) into a separate heat resistant bowl and create a slurry with the cornstarch, then whisk in 1/4 cup Parmesan. When the milk is fairly hot, pour a small amount of it into the slurry, stirring until the milk is thoroughly blended in. Return this to the remaining milk, add cognac, and simmer for 15 minutes, stirring frequently. Remove saucepan from heat, season sauce, to taste, with white pepper, and salt if needed. Remove and discard bay leaf, cloves and lobster shells. Immediately put a sheet of plastic wrap on top of the sauce to prevent a skin from forming.
Pre-heat oven to 350 degrees F. Gently fold the lobster and crabmeat together with just enough of the sauce to bind it. (Reserve the rest of the sauce for service.) If the filling seems loose, add more Parmesan to bind it. (You don't want the filling to be too runny.) Lay the cabbage leaves out and remove stiff ribs/spines of leaves being sure to leave enough of the leaf in a shape that can be filled and rolled shut. Into each leaf, spoon 3 or 4 tablespoons of the lobster/crabmeat mixture, add a few chunks of avocado, fold in sides of leaf, and roll up. Place in an oven-proof dish layered on the bottom with a little of the sauce, sprinkle with a little Parmesan. Bake for 10 minutes to melt cheese.
Serve garnished with fresh dill and with remaining lobster sauce on the side.