The advancement of face recognition technology has delivered substantial societal advantages. However, it has also raised global privacy concerns due to the ubiquitous collection and potential misuse of individuals' facial data. This presents a notable paradox: while there is a societal demand for a robust face recognition ecosystem to ensure public security and convenience, an increasing number of individuals are hesitant to release their facial data. Numerous studies have endeavored to find such a utility-privacy trade-off, yet many struggle with the dilemma of prioritizing one at the expense of the other. In response to this challenge, this paper proposes PRO-Face C, a novel paradigm for privacy-preserving recognition of obfuscated faces via a dedicated feature compensation mechanism, aimed at optimizing the equilibrium between privacy preservation and utility maximization. The proposed approach is characterized by a specialized client-server architecture: the client transmits only obfuscated images to the server, which then performs identity recognition using a pre-trained model in conjunction with a suite of privacy-free complementary features. This framework facilitates accurate face identification while safeguarding the original facial appearance from explicit disclosure. Furthermore, the obfuscated image retains its visualization capability, crucial for image preview functionalities. To ensure the desired properties, we have developed an identity-guided feature compensation mechanism, complemented by several privacy-enhancing techniques. Extensive experiments conducted across multiple face datasets underscore the effectiveness of the proposed approach in diverse scenarios.