Answer
Configure the Custom Event with an Event Action of type Application Message, selecting the specific fields you want included in the outbound message. Then create a new Outbound Routing Rule with Route From set to APPLICATION_MESSAGE on the relevant Queue, a Routing Condition matching the specific event's message function/ID, and a Routing Address of the appropriate connector type. For a genuine external REST/HTTP API, as opposed to a file-based staging step, configure that Routing Address as an HTTP-type Routing Address pointed directly at the third-party API's URL rather than a File Connector, adding a Transformer in between if the outbound XML the event produces needs to be reshaped into whatever specific payload structure the third-party API actually expects.