Does brother mfc-9340cdw works with macOS Monterey?

Hi,

A Brother MFC-9340CDW is listed in About AirPrint - Apple Support as a model supporting Apple's AirPrint technology. AirPrint is supported by macOS (12 Monterey). This means that the MFC-9340CDW should work without the installation of drivers, provided that the printer and the Mac are connected to the same network (Wi-Fi router).

AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections. Since the MFC-9340CDW has Ethernet (in addition to Wi-Fi), you could choose to wire the printer (Ethernet cable) to one of the LAN ports of the router. Ethernet may be more stable.

Add a printer to your printer list so you can use it on Mac – Apple Support (UK)

A USB connection would require drivers.

https://support.brother.com/g/b/downloadtop.aspx?c=us&lang=en&prod=mfc9340cdw_all

Indicates Catalina support is available. Which is good, because that's 64 bit. I would download the Catalina support, and see if it connects.

If it doesn't connect hold the Option key when adding the printer. This will bring up the open source drivers of any printer on WiFi or connected via USB to the computer.