As generative language models (GLMs) have gained popularity, youth are increasingly using them in their everyday lives. As such, most research has centered on supporting youth as users of GLM-powered systems. However, we know little of how to engage youth in the design of these models. Building on the rich legacy of child-computer interaction research that positions youth as designers of computing systems, we explore how to support young people in designing GLMs. Through a case study of three teenagers (ages 14-15) building a babyGPT screenplay generator, we illustrate how the team developed a model while engaging in artificial intelligence/machine learning-relevant data practices and addressing ethical issues. This paper contributes a case study that demonstrates the feasibility of engaging youth in building GLMs.