I'm not entirely sure what a specific 'big fat cock story' would be. It could potentially be a tall tale or an elaborate story related to a large rooster perhaps. Without more context, it's difficult to say exactly.
No, I don't think 'big fat cock story' is a common type of story. Most stories are about people, animals in general, or fictional characters, not specifically about a 'big fat cock' in this rather unusual context.
I'm not entirely sure what a specific 'fat cock story' would be about as the term 'fat cock' could be used in various ways in different contexts. It might refer to a story about a large rooster in a farmyard setting, for example, where the rooster has some unique characteristics or adventures.
There was this party where a black - feathered cockerel was brought in as a sort of novelty. A blonde lady was intrigued by it. She asked the owner if she could hold it. As she held the cockerel gently, she told everyone around about her childhood on a farm where they had similar birds. People gathered around her listening to her story, and the cockerel seemed quite content in her arms for a while. It made the party even more interesting.
Sharing such stories is highly inappropriate as it involves private and often vulgar sexual content. We should respect the boundaries of privacy and propriety in society.
There was this big fat tree in the middle of a forest. It was so huge that it could shelter dozens of animals during a storm. Birds made their nests on its branches, squirrels ran up and down its trunk. It was like a giant guardian of the forest. Many a time, lost hikers would find their way by using this big fat tree as a landmark.