  In this paper, we prove that for any asymptotically conical self-shrinker,
there exists an embedded closed hypersurface such that the mean curvature flow
starting from it develops a singularity modeled on the given shrinker. The main
technique is the Wa\.zewski box argument, used by Stolarski in the proof of the
corresponding theorem in the Ricci flow case. As a corollary, our construction,
combined with the works of Angenent-Ilmanen-Vel\'azquez and
Chodosh-Daniels-Holgate-Schulze, implies the existence of fattening level set
flows starting from smooth embedded closed hypersurfaces. This addresses a
question posed by Evans-Spruck and De Giorgi.
