Hi star
You can outsource the recruitment. Training is a different story though and has to be through a regulated company, ie BA, EK, etc. The actual contract of employment can be outsourced but you would need to determine how to obtain the licences for the crew (maximum three allowed) .The operation could be difficult depending on whether the agency or airline is managing matters like sickness, disruption and rostering.
An airline can use any agency to select suitable staff for them but it doesn't necessarily mean the agency will do a good job of it. They would need to select a reputable company, and would need a thorough contract, covering procurement requirements, numbers, fees, KPIs, remuneration, process, etc.
Cheers
Six