The answer is… You can recycle the lid… Only the metal end.
To recycle what you can from a frozen juice can, you just need to remove the metal lid from it. Remove the cardboad looking part from it, and recycle the lid. Then the cardboard goes to the garbages.
Why does the cardboard from frozen juice cans go to the garbage?
The cardboard is made of multiple layers of paper, plastic and foil to prevent the container from becoming soggy. Those multiple layers of materials cannot be easily separated for recycling.
So, put the cardboard in the garbage bin, and the clean lid on the recycling.