Is the Beckmann Rearrangement technically an "SN2" reaction at an "sp2" center?
When you were taught that we don't see $S_{\mathrm{N}}2$ reactions at $sp^{2}$ centers, what evidence or rationale was given? In organic chemistry, very few things are definite. When they are, it's usually because of being forbidden by quantum mechanics (e.g., symmetry in pericyclic reactions). Otherwise, they are just very unlikely, unless you somehow magically make the conditions right.

Greg Fu used to give this example: J. Am. Chem. Soc. 1998, 120, 2275-2282.

In it, a trans-substituted alkene undergoes nucleophilic substitution to become a cis-substituted alkene. Using that fact and other data, the authors conclude that the mechanism most consistent with the observed reaction data is a vinylic $S_{\mathrm{N}}2$. If you have a better mechanism, consider writing a paper and submitting it to JACS.