Few-shot dataset generalization is a challenging variant of the well-studied few-shot classification problem where a diverse training set of several datasets is given, for the purpose of training an adaptable model that can then learn classes from \emph{new datasets} using only a few examples. To this end, we propose to utilize the diverse training set to construct a \emph{universal template}: a partial model that can define a wide array of dataset-specialized models, by plugging in appropriate components. For each new few-shot classification problem, our approach therefore only requires inferring a small number of parameters to insert into the universal template. We design a separate network that produces an initialization of those parameters for each given task, and we then fine-tune its proposed initialization via a few steps of gradient descent. Our approach is more parameter-efficient, scalable and adaptable compared to previous methods, and achieves the state-of-the-art on the challenging Meta-Dataset benchmark.