Functional regression analysis is an established tool for many contemporary scientific applications. Regression problems involving large and complex data sets are ubiquitous, and feature selection is crucial for avoiding overfitting and achieving accurate predictions. We propose a new, flexible and ultra-efficient approach to perform feature selection in a sparse high dimensional function-on-function regression problem, and we show how to extend it to the scalar-on-function framework. Our method, called FAStEN, combines functional data, optimization, and machine learning techniques to perform feature selection and parameter estimation simultaneously. We exploit the properties of Functional Principal Components and the sparsity inherent to the Dual Augmented Lagrangian problem to significantly reduce computational cost, and we introduce an adaptive scheme to improve selection accuracy. In addition, we derive asymptotic oracle properties, which guarantee estimation and selection consistency for the proposed FAStEN estimator. Through an extensive simulation study, we benchmark our approach to the best existing competitors and demonstrate a massive gain in terms of CPU time and selection performance, without sacrificing the quality of the coefficients' estimation. The theoretical derivations and the simulation study provide a strong motivation for our approach. Finally, we present an application to brain fMRI data from the AOMIC PIOP1 study. Complete FAStEN code is provided at https://github.com/IBM/funGCN.