Recognizing and differentiating among both familiar and unfamiliar faces is a critical capability for face recognition systems and a key step toward artificial general intelligence (AGI). Motivated by this ability, this paper introduces generalized face discovery (GFD), a novel open-world face recognition task that unifies traditional face identification with generalized category discovery (GCD). GFD requires recognizing both labeled and unlabeled known identities (IDs) while simultaneously discovering new, previously unseen IDs. Unlike typical GCD settings, GFD poses unique challenges due to the high cardinality and fine-grained nature of face IDs, rendering existing GCD approaches ineffective. To tackle this problem, we propose FaceGCD, a method that dynamically constructs instance-specific feature extractors using lightweight, layer-wise prefixes. These prefixes are generated on the fly by a HyperNetwork, which adaptively outputs a set of prefix generators conditioned on each input image. This dynamic design enables FaceGCD to capture subtle identity-specific cues without relying on high-capacity static models. Extensive experiments demonstrate that FaceGCD significantly outperforms existing GCD methods and a strong face recognition baseline, ArcFace, achieving state-of-the-art results on the GFD task and advancing toward open-world face recognition.