A number of applications (e.g., video surveillance and authentication) rely on automated face recognition to guarantee functioning of secure services, and meanwhile, have to take into account the privacy of individuals exposed under camera systems. This is the so-called Privacy-Utility trade-off. However, most existing approaches to facial privacy protection focus on removing identifiable visual information from images, leaving protected face unrecognizable to machine, which sacrifice utility for privacy. To tackle the privacy-utility challenge, we propose a novel, generic, effective, yet lightweight framework for Privacy-preserving Recognizable Obfuscation of Face images (named as PRO-Face). The framework allows one to first process a face image using any preferred obfuscation, such as image blur, pixelate and face morphing. It then leverages a Siamese network to fuse the original image with its obfuscated form, generating the final protected image visually similar to the obfuscated one from human perception (for privacy) but still recognized as the original identity by machine (for utility). The framework supports various obfuscations for facial anonymization. The face recognition can be performed accurately not only across anonymized images but also between plain and anonymized ones, based on only pre-trained recognizers. Those feature the "generic" merit of the proposed framework. In-depth objective and subjective evaluations demonstrate the effectiveness of the proposed framework in both privacy protection and utility preservation under distinct scenarios. Our source code, models and any supplementary materials are made publicly available.