Bug tracking enables the monitoring and resolution of issues and bugs within organizations. Bug triaging, or assigning bugs to the owner(s) who will resolve them, is a critical component of this process because there are many incorrect assignments that waste developer time and reduce bug resolution throughput. In this work, we explore the use of a novel two-output deep neural network architecture (Dual DNN) for triaging a bug to both an individual team and developer, simultaneously. Dual DNN leverages this simultaneous prediction by exploiting its own guess of the team classes to aid in developer assignment. A multi-label classification approach is used for each of the two outputs to learn from all interim owners, not just the last one who closed the bug. We make use of a heuristic combination of the interim owners (owner-importance-weighted labeling) which is converted into a probability mass function (pmf). We employ a two-stage learning scheme, whereby the team portion of the model is trained first and then held static to train the team--developer and bug--developer relationships. The scheme employed to encode the team--developer relationships is based on an organizational chart (org chart), which renders the model robust to organizational changes as it can adapt to role changes within an organization. There is an observed average lift (with respect to both team and developer assignment) of 13%-points in 11-fold incremental-learning cross-validation (IL-CV) accuracy for Dual DNN utilizing owner-weighted labels compared with the traditional multi-class classification approach. Furthermore, Dual DNN with owner-weighted labels achieves average 11-fold IL-CV accuracies of 76% (team assignment) and 55% (developer assignment), outperforming reference models by 14%- and 25%-points, respectively, on a proprietary dataset with 236,865 entries.