Advancements in Automated Driving Systems (ADSs) have enabled the achievement of a certain level of autonomy while commuting in a car. However, emergency and high-speed maneuvers still arise as significant challenges for ADSs due to the intrinsic nonlinearity and fast-paced behavior of such events. These maneuvers are a distinctive feature within the recently established motorsport discipline of Autonomous Racing (AR). In this work, we explore the use of Learning-based Model Predictive Control (LMPC) to address possible model mismatches of the first principles model in high-speed racing. To this end, a Model Predictive Contouring Control (MPCC) (a specific formulation of the standard Model Predictive Control, MPC) is formulated, and a Neural Network (NN) that leverages the use of Feedforward and Recurrent layers is employed to learn the errors of the first principles model. By combining the NN with the first principles model, the LMPC is born, capable of accurately predicting the future with a computational effort compatible with real-time feasibility, effectively handling the vehicle at its limits. Furthermore, the controller can adapt to changing environments by training the NN during the race. The MPCC (formulation without the NN) is deployed on a real autonomous formula student racing car showing an improvement of 16 % in mean lap times across the same track between a common geometric controller. The LMPC is analyzed in a high-fidelity simulator, achieving an improvement of 8.9 % in mean lap times when compared to the MPCC.