We show that for a d-dimensional model in which a quench with a rate tau(-1) takes the system across a (d-m)-dimensional critical surface, the defect density scales as n similar to 1/tau(m nu/(z nu+1)), where nu and z are the correlation length and dynamical critical exponents characterizing the critical surface. We explicitly demonstrate that the Kitaev model provides an example of such a scaling with d=2 and m=nu=z=1. We also provide the first example of an exact calculation of some multispin correlation functions for a two-dimensional model that can be used to determine the correlation between the defects. We suggest possible experiments to test our theory.