So you have a BX but the fans are a bit too loud for you. Well I have some good news and some bad news.
The good news. There are ways to make the fans more quiet and in fact you can make them completely silent by turning them off when they are not needed. The bad news is that when the fans are needed you cannot avoid at least some noise.
Most of the noise from a fan is generated by rushing air and vibrations against the chassis of the machine. We will deal with the rushing air in the next part but for now let's talk about vibrations.
In the image below you can see the primary guilty party, the 5020 blower fan within the electronics enclosure. It is used to cool the 400MHz CPU as well as the TMC drivers. You want it on when the machine is doing serious number crunching and driving the steppers but you don't want it to vibrate against the enclosure and amplify the noise. A common trick is to simply place a rubber washer in between the enclosure and the fan which dampens the vibrations. As you can see in the picture you will be able to access the one screw from above but the other will require you to drop the electronics enclosure out from below. Pop the washers in and your fan will be a lot more quiet.
Once a print is done, all of the complex number crunching and motor driving that generate heat stop. There is therefore no need to be cooling the internal electronics and the fans can be turned off. Once another print starts or when the motors start to move again, you will need the fans back on.
This modification makes the fans behave in that exact manner and it is really easy to do. Simply unplug the internal fans from the two connectors that are circled in red in the picture below. The top connector belongs to the blower fan and the bottom connector (white one hiding behind the black wire) belongs to the front fan with the super bright blue LED. There may be some glue on the connectors so you'll have to pull that off first. Some people have mentioned that they are difficult to get to and that is a valid point. I simply used tweezers to grip the plugs on either side and pull them out (obviously with the power off).
Once the connectors are off you need to connect them to the new plugs on the PCB identified in the picture below. Connect the blower fan to the plug on the left and the front case fan to the plug on the right.
Once the fans are in place they will now be controllable by the software. Install the software located at the following link which will control them. The blower fan will come on whenever the motors are moving and the front case fan will come on whenever the part cooling fan is on during a print.
I ordered a new one in the end. I was told that it was a fairly common occurence on BTT mainboards by the reseller. He apparently broke one the exact same way.