This paper presents an original approach to vehicle obstacle avoidance. It involves the development of a nonlinear Model Predictive Contouring Control, which uses torque vectoring to stabilise and drive the vehicle in evasive manoeuvres at the limit of handling. The proposed algorithm combines motion planning, path tracking and vehicle stability objectives, prioritising collision avoidance in emergencies. The controller's prediction model is a nonlinear double-track vehicle model based on an extended Fiala tyre to capture the nonlinear coupled longitudinal and lateral dynamics. The controller computes the optimal steering angle and the longitudinal forces per each of the four wheels to minimise tracking error in safe situations and maximise the vehicle-to-obstacle distance in emergencies. Thanks to the optimisation of the longitudinal tyre forces, the proposed controller can produce an extra yaw moment, increasing the vehicle's lateral agility to avoid obstacles while keeping the vehicle stable. The optimal forces are constrained in the tyre friction circle not to exceed the tyres and vehicle capabilities. In a high-fidelity simulation environment, we demonstrate the benefits of torque vectoring, showing that our proposed approach is capable of successfully avoiding obstacles and keeping the vehicle stable while driving a double-lane change manoeuvre, in comparison to baselines lacking torque vectoring or collision avoidance prioritisation.