One of the first steps of cheese making is to suppress the colloidal stability of casein micelles by enzymatic hydrolysis and initiate milk gelation. Afterwards, the enzymatic milk gel is cut to promote syneresis and expulsion of the soluble phase of milk. Many studies have reported on the rheological properties of enzymatic milk gels at small strain, but they provide limited information on the ability of the gel to be cut and handled. In this study, we aim to characterize the non-linear properties and the yielding behavior of enzymatic milk gels during creep, fatigue and stress sweep tests. We evidence by both continuous and oscillatory shear tests that enzymatic milk gel displays irreversible and brittle-like failure, as reported for acid caseinate gels, but with additional dissipation during fracture opening. Before yielding, acid caseinate gels display strain-hardening only, while enzymatic milk gels also display strain-softening. By varying the gel aging time and the volume fraction of casein micelles, we are able to attribute the hardening to the network structure and the softening to local interactions between casein micelles. Our study highlights the crucial importance of the nanoscale organization of the casein micelles - or more generally of the building block of a gel - to retain the macroscopic nonlinear mechanical properties of the gel.