Spectral clustering (SC) is a popular graph clustering technique to find strongly connected communities on a graph, which can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is an expensive operation and, since clustering results are graph-specific, pooling methods based on SC cannot generalize to unseen samples. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem and train a GNN to compute cluster assignments that minimize this objective. Our GNN-based implementation is differentiable, does not require to compute the spectral decomposition, and allows to cluster out-of-sample graphs. From the proposed clustering method, we design a graph pooling operator that overcomes some important limitations of state-of-the-art graph pooling techniques and achieves better performance in several supervised and unsupervised tasks.