Neural networks (NNs) and graph signal processing have emerged as important actors in data-science applications dealing with complex (non-linear, non-Euclidean) datasets. In this work, we introduce a novel graph-aware NN architecture to learn the mapping between graph signals that are defined on two different graph datasets. The novel proposed architecture is based on two NNs and a common latent space. In particular, we consider an underparametrized graph-aware NN encoder that maps the input graph signal to a latent space, followed by an underparametrized graph-aware NN decoder which maps the latent representation to the output graph signal. The parameters of the two NN are jointly learned by using a training set and the back-propagation algorithm. The resulting architecture can then be viewed as an underparametrized graph-aware encoder/decoder NN operating over two different graphs. The proposed scheme outperforms the corresponding benchmark NN architectures in the literature.