While Genetic Improvement (GI) is a useful paradigm to improve functional and nonfunctional aspects of software, existing techniques tended to use the same set of mutation operators for differing objectives, due to the difficulty of writing custom mutation operators. In this work, we suggest that Large Language Models (LLMs) can be used to generate objective-tailored mutants, expanding the possibilities of software optimizations that GI can perform. We further argue that LLMs and the GI process can benefit from the strengths of one another, and present a simple example demonstrating that LLMs can both improve the effectiveness of the GI optimization process, while also benefiting from the evaluation steps of GI. As a result, we believe that the combination of LLMs and GI has the capability to significantly aid developers in optimizing their software.