Cooking is leveled up by simply cooking. In order to be able to cook you have to camp at Havens. Resting anywhere else will not let you cook anything.
The best way i found to grind this out is to go fish up a bunch of trout and buy or gather Leiden peppers. You can make Skewered Wild Trout with those. Just sit at a Haven and keep camping and cooking that. It took about 10 to get from level 9 to 10.