Cutting planes are central to mixed-integer linear programming (MILP) solving, yet their effectiveness hinges on expert tuning of separator configurations and hard-crafted cut-selection heuristics, creating a high barrier for non-specialists. Learning-based methods can reduce manual effort, but typically require large training datasets and often generalize poorly beyond the instance classes they are trained on. We propose an LLM-guided cutting-plane management framework that integrates large language models into the MILP solving pipeline. First, using chain-of-thought (CoT) prompting, the LLM infers an instance-specific separator configuration, deciding which separators to activate and how to set key parameters from the problem type and structural features. Then, it translates the evolving branch-and-bound state in natural language to perform stage-aware cut selection, choosing a high-quality subset of cuts that tightens the relaxation and improves overall solver performance. Leveraging the LLM's reasoning capabilities and rich background knowledge, our work removes dependence on domain-specific training data and substantially reduces reliance on expert-crafted configurations. Experiments show consistent improvements over SCIP's default settings, hard-crafted heuristics, and recent learning-based cut selection baselines.