Congestion is a critical and challenging problem in communication networks. Congestion control protocols allow network applications to tune their sending rate in a way that optimizes their performance and the network utilization. In the common distributed setting, the applications cannot collaborate with each other directly but instead obtain similar estimations about the state of the network using latency and loss measurements. These measurements can be fed into analytical functions, referred to by utility functions, whose gradients help each and all distributed senders to converge to a desired state. The above process becomes extremely complicated when each application has different optimization goals and requirements. Crafting these utilization functions has been a research subject for over a decade, with small incremental changes requiring rigorous mathematical analysis as well as real-world experiments. In this work, we present GenCC, a framework leveraging the code generation capabilities of large language models (LLMs) coupled with realistic network testbed, to design congestion control utility functions. Using GenCC, we analyze the impact of different guidance strategies on the performance of the generated protocols, considering application-specific requirements and network capacity. Our results show that LLMs, guided by either a generative code evolution strategy or mathematical chain-of-thought (CoT), can obtain close to optimal results, improving state-of-the-art congestion control protocols by 37%-142%, depending on the scenario.