This paper addresses the problem of modeling loss frequency using regression when the counts have a non-standard distribution. We propose a novel approach based on mixture-of-experts specifications on discrete-phase type distributions. Compared to continuous phase-type counterparts, our approach offers fast estimation via expectation-maximization, making it more feasible for use in real-life scenarios. Our model is both robust and interpretable in terms of risk classes, and can be naturally extended to the multivariate case through two different constructions. This avoids the need for ad-hoc multivariate claim count modeling. Overall, our approach provides a more effective solution for modeling loss frequency in non-standard situations.