This research establishes a novel combined plant and controller optimization (termed co-design) framework, aimed at complex systems, that speeds up the optimization process by adjusting controller parameters during an experiment while plant parameters are adjusted between batches of experiments. Most legacy co-design approaches have been restricted to analytical or numerical approaches that require full knowledge of the system dynamics and do not leverage the unique ability to optimize control parameters in real time. To address these challenges, the proposed nested co-design framework relies on an iterative outer loop to adjust the plant design and an inner loop that optimizes the control parameters during an experiment. Following each round of experiments, performance of the dynamical system is characterized across the design space, along with a statistical characterization of uncertainty. Using these characterizations, the design space is reduced prior to proceeding to subsequent iterations. The process is repeated until the design space has been sufficiently reduced. This dissertation evaluates a variety of candidate methodologies for both the outer-loop plant iteration and inner-loop control adaptation, including statistical design of experiments, Gaussian Process (GP) modeling, and more traditional adaptive control techniques. The dissertation focuses heavily on the fusion of a GP-based technique for plant iteration and a novel recursive GP (RGP)-based adaptive control technique for control parameter adaptation. Not only does this result in a formulation where the inner-loop control parameter adaptation mirrors the iteration-based plant parameter adaptation; the RGP-based adaptation technique also represents a standalone contribution to the adaptive control literature. All variations of this framework have been validated in simulations and/or lab-scale experiments for an airborne wind energy system. The results in this dissertation demonstrate the efficacy of the nested co-design framework in efficiently converging to optimal design parameters.