Can start with simple tests like moisture content, fat content, oxidative rancidity test, color determination, texture analysis, even volume check etc from the point of frying to 1 month or until the longest period of storage time…
Few sulfides are present in fresh onion: methyl propyl disulfide, dipropyl disulfide and dipropyl trisulfide. The quantity of these sulphide might reduce (possibility to check and relate to the flavour & quality of the stored product).
Certain aldehydes are present in raw onion. eg, 2-methyl 2-pentenal. How about quantifying its amount or any newly formed aldehyde after frying and storage (such as 2-methylbutanal and 3-methylbutanal and correlate to frying temperature & storage period).
Sensory attributes are good to check.
(not sure if you stored this in the powder format or stored as fried onion itself?).