JavaScript engines are widely used in web browsers, PDF readers, and server-side applications. The rise in concern over their security has led to the development of several targeted fuzzing techniques. However, existing approaches use random selection to determine where to perform mutations in JavaScript code. We postulate that the problem of selecting better mutation targets is suitable for combinatorial bandits with a volatile number of arms. Thus, we propose CLUTCH, a novel deep combinatorial bandit that can observe variable length JavaScript test case representations, using an attention mechanism from deep learning. Furthermore, using Concrete Dropout, CLUTCH can dynamically adapt its exploration. We show that CLUTCH increases efficiency in JavaScript fuzzing compared to three state-of-the-art solutions by increasing the number of valid test cases and coverage-per-testcase by, respectively, 20.3% and 8.9% on average. In volatile and combinatorial settings we show that CLUTCH outperforms state-of-the-art bandits, achieving at least 78.1% and 4.1% less regret in volatile and combinatorial settings, respectively.