Exactly as above - the cooking oil itself would only be involved as a secondary fire and would require a well established primary fire to become involved due to the high flash point and auto ignition temperature.
The oil would be a fire service problem to deal with, which would require most probably a continuous attack with aspirated foam in quantity - completely different from the 6 or 9 litres of non aspirated foam available in extinguishers, hence why fryers need wet chemical.
You should look at the types of primary ignition & fuel sources and provide an appropriate agent - CO2 for enclosed electrical ignition sources, foam or powder for mixed A/B fuels.
Powder - rapid knockdown, but no cooling (reignition risk), must totally extinguish liquids in one go or you get flashback, and of course obscures vision and makes a mess. No reignition potential. Works in all UK temperatures so can be placed in outside storage compounds
Foam - slower extinguishing, but cools, allows partial or progressive extinguishing of liquids, less mess. Foam blanket prevets reignition & can be used on spills to protect against a fire in the first place prior to clean up.Requires low freeze additive if kept outside in parts of the UK
The ultimate is to use the 'two by four' approach as per UK motorsport - powder for rapid knockdown then foam to cool & protect.