They used to offer immediate processing for those that appeared in person at the Licensing Division who met a heightened criteria. It had to be an extreme need (like you just became the state's star witness against the mob on live tv), but you'd leave with a temporary permit that day for use until your real permit was processed. If you didn't qualify for the temporary permit, they just took your application and added it to the pile.
They also processed any police officer who appeared at the building under the same timeline, but they always left with the temporary permit.
I'm not sure if those pathways are still offered, they were the policy in 2004ish.
They are indeed still available, however, before they hand you your W&C permit, you need to submit training. There are several members on here that have received a W&C permit in less than a week, based on immediate need.