Yes, it would, though a hard fat such as lard would last a long time in cool conditions. The topping would probably be discarded anyway. This is really a method of preservation that works long enough to get food to market where transport is slow. Another, which avoids having to use a dish, is to cook the food in a large pie made of hard flour-and-water 'huff paste', which again would be discarded or thrown to the dogs. For example, the lampreys that were an expensive delicacy in medieval England (and of which King Henry I died of a surfeit), and which are highly perishable, were transported inland in pies.

RH