Large Language Models (LLMs) excel at countless tasks, yet struggle with creativity. In this paper, we introduce a novel approach that couples LLMs with structured representations and cognitively inspired manipulations to generate more creative and diverse ideas. Our notion of creativity goes beyond superficial token-level variations; rather, we explicitly recombine structured representations of existing ideas, allowing our algorithm to effectively explore the more abstract landscape of ideas. We demonstrate our approach in the culinary domain with DishCOVER, a model that generates creative recipes. Experiments comparing our model's results to those of GPT-4o show greater diversity. Domain expert evaluations reveal that our outputs, which are mostly coherent and feasible culinary creations, significantly surpass GPT-4o in terms of novelty, thus outperforming it in creative generation. We hope our work inspires further research into structured creativity in AI.