We consider coded caching over the fading broadcast channel, where the users, equipped with a memory of finite size, experience asymmetric fading statistics. It is known that a naive application of coded caching over the channel at hand performs poorly especially in the regime of a large number of users due to a vanishing multicast rate. We overcome this detrimental effect by a careful design of opportunistic scheduling policies such that some utility function of the long-term average rates should be maximized while balancing fairness among users. In particular, we propose a threshold-based scheduling that requires only statistical channel state information and one-bit feedback from each user. More specifically, each user indicates via feedback whenever its SNR is above a threshold determined solely by the fading statistics and the fairness requirement. Surprisingly, we prove that this simple scheme achieves the optimal utility in the regime of a large number of users. Numerical examples show that our proposed scheme performs closely to the scheduling with full channel state information, but at a significantly reduced complexity.