Large observational studies derived from electronic health record (EHR) data are increasingly being used for comparative effectiveness research. Though these data have many advantages, investigators must acknowledge and handle a typically substantial amount of missing data. Most existing methods for missing data focus on identification and estimation of parameters of interest when data are missing at random, however this assumption is likely untenable in EHR data for which the missingness process is complex and poorly understood. We consider a double sampling design in which a subsample of subjects with initially missing data are more intensively followed up to obtain complete information. We discuss scenarios and assumptions under which the joint density of interest is identified in the augmented sample. Further, we present semiparametric efficient and multiply robust estimators of causal average treatment effects when outcome data are initially missing not at random. Finally, we demonstrate our statistical approach, as well as the practical feasibility of the design, in an EHR-based analysis of weight outcomes following bariatric surgery.