We present new analytic techniques for inferring HTTP semantics from passive observations of HTTPS that can infer the value of important fields including the status-code, Content-Type, and Server, and the presence or absence of several additional HTTP header fields, e.g., Cookie and Referer. Our goals are twofold: to better understand the limitations of the confidentiality of HTTPS, and to explore benign uses of traffic analysis such as application troubleshooting and malware detection that could replace HTTPS interception and static private keys in some scenarios. We found that our techniques improve the efficacy of malware detection, but they do not enable more powerful website fingerprinting attacks against Tor. Our broader set of results raises concerns about the confidentiality goals of TLS relative to a user's expectation of privacy, warranting future research. We apply our methods to the semantics of both HTTP/1.1 and HTTP/2 on data collected from automated runs of Firefox 58.0, Chrome 63.0, and Tor Browser 7.0.11 in a lab setting, and from applications running in a malware sandbox. We obtain ground truth plaintext for a diverse set of applications from the malware sandbox by extracting the key material needed for decryption from RAM post-execution. We developed an iterative approach to simultaneously solve several multi-class (field values) and binary (field presence) classification problems, and we show that our inference algorithm achieves an unweighted $F_1$ score greater than 0.900 for most HTTP fields examined.