In this talk, I will illustrate the design and the a priori analysis of a new nonconforming finite element method for second-order elliptic problems, based on the first-order Crouzeix-Raviart space. The main feature of the proposed method is that it is quasi-optimal, i.e. its energy-norm error is proportional to the best approximation error. Quasi-optimality is known to hold for conforming finite element methods, according to the so-called Céa's lemma. In contrast, a simple argument actually reveals that several existing nonconforming methods do not enjoy such property. To introduce the announced construction, I will additionally motivate the use of nonconforming and quasi-optimal methods. Possible generalizations will also be addressed. This is a joint work with Andreas Veeser.