Going by known pairs of nouns and verbs, it would be "cognite", but is it valid? If not, what's the right word?
Update: I know cognise or cognize, but it doesn't cover the full meaning of "cognition" as it's used now. E.g., AHD has "The mental process or faculty of knowing, including aspects such as awareness, perception, reasoning, and judgment." Consider the usage in "Cognitive-Behavorial Therapy". It's far more than just the meaning of "know".
Furthermore, cognize is the verb for cognizance, not cognition.