This paper presents a new model-free methodology to learn Kazantzis-Kravaris-Luenberger (KKL) observers for nonlinear systems. We address three major difficulties arising in observer design: the peaking phenomenon, the noise sensitivity and the trade-off between convergence speed and robustness. We formulate the learning objective as an optimization problem, strictly minimizing the error of the observer estimates, without the need of adding explicit constraints or regularization terms. We further improve the performance with a switching approach, efficiently transitioning between two observers, respectively designed for the transient phase and the asymptotic convergence. Numerical results on the Van der Pol system, the Rössler attractor and on a bioreactor illustrate the gain of the method regarding the literature, in term of performance and robustness. Code available online: https://github.com/jolindien-git/DeepKKL