Cut butter or margarine into six pieces. Place water and butter in a medium saucepan. Bring to a full boil over medium heat until butter is melted.
Add flour and salt all at once. Stir vigorously until the dough forms a ball and leaves the side of the pan, about one minute. Remove from heat. Let stand five minutes.
Add eggs, one at a time, mixing thoroughly after each addition. After the last egg is mixed in, mix vigorously 30 seconds longer.
Stir in shrimp, green onions and four tbsp Parmesan.
Drop shrimp dough by heaping spoonfuls about two inches apart on baking sheets lined with parchment paper. Sprinkle with remaining Parmesan cheese. Bake 20 to 30 minutes or until puffed and golden brown.