How do I get power into my kitchen cabinet?
I'd use the range hood box as a junction box.  It meets the code definition of accessible:

Capable of being removed or exposed without damaging the building structure or finish or not permanently closed in by the structure or finish of the building.

And in most places should pass just fine.  I'd use BX because that MDF hood cover is rather easily removable, and treat the wiring to the new outlet as 'exposed'.