We consider the problem of co-designing embodied intelligence as a whole in a structured way, from hardware components such as propulsion systems and sensors to software modules such as control and perception pipelines. We propose a principled approach to formulate and solve complex embodied intelligence co-design problems, leveraging a monotone co-design theory. The methods we propose are intuitive and integrate heterogeneous engineering disciplines, allowing analytical and simulation-based modeling techniques and enabling interdisciplinarity. We illustrate through a case study how, given a set of desired behaviors, our framework is able to compute Pareto efficient solutions for the entire hardware and software stack of a self-driving vehicle.