Since updating to Mac OS11 Big Sur, I have had long delays in printing on my Brother printer. The Printer Queue goes into "looking for printer"; I do eventually get a printed page but only after I have gotten tired of waiting and left the room. Went to Brother support and there is no Driver for Mac OS 11 to date. One work around is I send the document to my email then print from my iPhone or iPad using the mobile app from Brother. On a separate issue, after updating to Big Sur, my Mac mini 2018 has frozen and rebooted more times in the last few days than in the last 5 years of using Mac's. Fortunately, the crashes are being sent to Apple.

I had a similar problem with my Brother Printer (AirPrint capable) after installing Mac OS 11 Big Sur. My printer was found in System Preferences>Printers & Scanners. But, when I hit print on a document the printer queue would go into terminal search for my printer. So, as per apple: "On your Mac, choose Apple menu > System Preferences, then click Printers & Scanners. Press and hold the Control key as you click in the list at the left, then choose “Reset printing system” from the menu that appears. After you reset the printing system, the list of printers in Printers & Scanners preferences is empty." Yet, after doing this no printer showed in the Printer & Scanners window. I unplugged my printer, waited x 10 sec and replugged in. My printer showed up under a Bonjour printer and now able to print again. Hope this helps. BTW Brother Printers still has no new driver for Mac OS 11 for my printer and recommends using AirPrint. Apple has a list of which printers are AirPrint capable: https://support.apple.com/en-us/HT201311