(Disclaimer: I not only welcome but invite clarification from actual Muslims on the following.)

I’m given to understand that you’re also supposed to say “Bismillah” when you sic a Saluki (the one specifically halal dog breed) on a prey animal; this authorizes the dog as a tool of halal slaughter - as long as they bring you the rabbit/gazelle/whatever without themself partaking, meaning that they were hunting on your behalf rather than their own.