Bayesian optimization is a framework for optimizing functions that are costly or time-consuming to evaluate. Recent work has considered Bayesian optimization of function networks (BOFN), where the objective function is represented by a network of functions, each taking as input the output of previous nodes in the network as well as additional parameters. Exploiting this network structure in the optimization has been shown to yield significant performance improvements. Existing BOFN algorithms for general-purpose networks evaluate the full network at each iteration. However, many real-world applications allow for evaluating nodes individually. To take advantage of this opportunity, we propose a novel knowledge gradient acquisition function for BOFN that chooses which node to evaluate as well as the inputs for that node in a cost-aware fashion. This approach can dramatically reduce query costs by evaluating only parts of the network. We provide an efficient approach to optimizing our acquisition function and show it outperforms existing BOFN methods and other benchmarks across several synthetic and real-world problems. Our acquisition function is the first to enable cost-aware optimization of a broad class of function networks.