By leveraging information technologies, organizations now have the ability to design their communication networks and crowdsourcing platforms to pursue various performance goals, but existing research on network design does not account for the specific features of social networks, such as the notion of teams. We fill this gap by demonstrating how desirable aspects of organizational structure can be mapped parsimoniously onto the spectrum of the graph Laplacian allowing the specification of structural objectives and build on recent advances in non-convex programming to optimize them. This design framework is general, but we focus here on the problem of creating graphs that balance high modularity and low mixing time, and show how "liaisons" rather than brokers maximize this objective.