Large language models (LLMs) have taken the scientific world by storm, changing the landscape of natural language processing and human-computer interaction. These powerful tools can answer complex questions and, surprisingly, perform challenging creative tasks (e.g., generate code and applications to solve problems, write stories, pieces of music, etc.). In this paper, we present a collaborative game design framework that combines interactive evolution and large language models to simulate the typical human design process. We use the former to exploit users' feedback for selecting the most promising ideas and large language models for a very complex creative task - the recombination and variation of ideas. In our framework, the process starts with a brief and a set of candidate designs, either generated using a language model or proposed by the users. Next, users collaborate on the design process by providing feedback to an interactive genetic algorithm that selects, recombines, and mutates the most promising designs. We evaluated our framework on three game design tasks with human designers who collaborated remotely.