Large Language Models (LLMs) are increasingly used to generate and shape cultural content, ranging from narrative writing to artistic production. While these models demonstrate impressive fluency and generative capacity, prior work has shown that they also exhibit systematic cultural biases, raising concerns about stereotyping, homogenization, and the erasure of culturally specific forms of expression. Understanding whether LLMs can meaningfully align with diverse cultures beyond the dominant ones remains a critical challenge. In this paper, we study cultural adaptation in LLMs through the lens of cooking recipes, a domain in which culture, tradition, and creativity are tightly intertwined. We build on the \textit{GlobalFusion} dataset, which pairs human recipes from different countries according to established measures of cultural distance. Using the same country pairs, we generate culturally adapted recipes with multiple LLMs, enabling a direct comparison between human and LLM behavior in cross-cultural content creation. Our analysis shows that LLMs fail to produce culturally representative adaptations. Unlike humans, the divergence of their generated recipes does not correlate with cultural distance. We further provide explanations for this gap. We show that cultural information is weakly preserved in internal model representations, that models inflate novelty in their production by misunderstanding notions such as creativity and tradition, and that they fail to identify adaptation with its associated countries and to ground it in culturally salient elements such as ingredients. These findings highlight fundamental limitations of current LLMs for culturally oriented generation and have important implications for their use in culturally sensitive applications.