A cold-stimulus headache is thought to be the direct result of the rapid cooling and rewarming of the capillaries in the sinuses leading to periods of vasoconstriction and vasodilation. A similar, but painless, blood vessel response causes the face to appear “flushed” after being outside on a cold day. In both instances, the low temperature causes the capillaries in the sinuses to constrict and then experience extreme rebound dilation as they warm up again.[13]

In the palate, this dilation is sensed by nearby pain receptors, which then send signals back to the brain via the trigeminal nerve, one of the major nerves of the facial area. This nerve also senses facial pain, so as the neural signals are conducted the brain interprets the pain as coming from the forehead—the same “referred pain” phenomenon seen in heart attacks. Brain-freeze pain may last from a few seconds to a few minutes. Research suggests that the same vascular mechanism and nerve implicated in “brain freeze” cause the aura (sensory disturbance) and pulsatile (throbbing pain) phases of migraines.[14]

https://en.wikipedia.org/wiki/Cold-stimulus_headache

I think it is the cooling of the top of the mouth, not the teeth.