Cellular network infrastructure serves as the backbone of modern mobile wireless communication. As such, cellular cores must be proactively secured against external threats to ensure reliable service. Compromised base station attacks against the core are a rising threat to cellular networks, while user device inputs have long been considered as an attack vector; despite this, few techniques exist to comprehensively test RAN-Core interfaces against malicious input. In this work, we devise a fuzzing framework that performantly fuzzes cellular interfaces accessible from a base station or user device, overcoming several challenges in fuzzing specific to LTE/5G network components. We also introduce ASNFuzzGen, a tool that compiles ASN.1 specifications into structure-aware fuzzing modules, thereby facilitating effective fuzzing exploration of complex cellular protocols. We run fuzzing campaigns against seven open-source and commercial cores and discover 119 vulnerabilities, with 93 CVEs assigned. Our results reveal common implementation mistakes across several cores that lead to vulnerabilities, and the successful coordination of patches for these vulnerabilities across several vendors demonstrates the practical impact ASNFuzzGen has on hardening user-exposed cellular systems.