We propose a novel Federated Edge Network Utility Maximization (FEdg-NUM) architecture for solving a large-scale distributed network utility maximization (NUM) problem. In FEdg-NUM, clients with private utilities communicate to a peer-to-peer network of edge servers. This represents a departure from the classical distributed NUM master-slave configuration and enables distributed computing harnessing local communications. Compared to a solution using cloud synchronization via Ring AllReduce, we prove that our federated edge computing model has shorter run-time in the presence of network congestion, thanks to its configuration and its ability to make progress in the presence of intermittent links. The paper studies its convergence and run-time performance both analytically and numerically, and illustrates several possible networking applications.