Traffic signal control TSC requires strategies that are both effective and interpretable for deployment, yet reinforcement learning produces opaque neural policies while program synthesis depends on restrictive domain-specific languages. We present SIGNALCLAW, a framework that uses large language models LLMs as evolutionary skill generators to synthesize and refine interpretable control skills for adaptive TSC. Each skill includes rationale, selection guidance, and executable code, making policies human-inspectable and self-documenting. At each generation, evolution signals from simulation metrics such as queue percentiles, delay trends, and stagnation are translated into natural language feedback to guide improvement. SignalClaw also introduces event-driven compositional evolution: an event detector identifies emergency vehicles, transit priority, incidents, and congestion via TraCI, and a priority dispatcher selects specialized skills. Each skill is evolved independently, and a priority chain enables runtime composition without retraining. We evaluate SignalClaw on routine and event-injected SUMO scenarios against four baselines. On routine scenarios, it achieves average delay of 7.8 to 9.2 seconds, within 3 to 10 percent of the best method, with low variance across random seeds. Under event scenarios, it yields the lowest emergency delay 11.2 to 18.5 seconds versus 42.3 to 72.3 for MaxPressure and 78.5 to 95.3 for DQN, and the lowest transit person delay 9.8 to 11.5 seconds versus 38.7 to 45.2 for MaxPressure. In mixed events, the dispatcher composes skills effectively while maintaining stable overall delay. The evolved skills progress from simple linear rules to conditional strategies with multi-feature interactions, while remaining fully interpretable and directly modifiable by traffic engineers.