Convert Extension methods to generic methods
Have you considered using in Interface on MyObject?

public static IQueryable&lt;T&gt; JoinWhere(this IQueryable&lt;T&gt; initialObject, IQueryable&lt;TJoinOn&gt; joinOn, IEnumerable&lt;string&gt; types) 
where T : IMyObject
{
    return (from myObject in initialObject
                                               //Here                      //Here
        join myObjectResourceType in joinOn on myObject.ResourceKey equals myObjectResourceType.ResourceKey
                             //Here
        where types.Contains(myObjectResourceType.ResourceType)
        select myObject);
}