How to expose a public API from a directive that is a reusable component?

Does this work for you?

angular.directive('extLabel', function() {
    return {
        restrict: 'E',
        scope: {
            api: '='
        },
        link: function(scope, iElement, iAttrs) {
            scope.api = {
                    doSomething: function() { },
                    doMore: function() { }
                };
        }
    };
});

From containing parent

<ext:label api="myCoolApi"></ext:label>

And in controller

$scope.myCoolApi.doSomething();
$scope.myCoolApi.doMore();