How to install MacOS on a MacBook with a broken screen ?

You need to enable Clamshell Mode so that only the external display is used. To enable Clamshell mode you need to connect the charger and a USB keyboard and mouse. Power on the laptop and immediately close the lid of the Display while also pressing the keys to enter Recovery Mode or Internet Recovery Mode (you may need to wait and do this immediately after the startup chime). It can be difficult to force video to the external display when the internal display is still "seen" by the system.

If you cannot get the timing correct, then try entering Recovery Mode or Internet Recovery Mode first and the laptop time to boot to the recovery utilities screen if you have connected the laptop directly to the router using an Ethernet cable and a USB-C to Gigabit Ethernet Adapter so you don't have to worry about entering WiFi credentials to access the Internet as getting an external display to work at that stage is extremely difficult and it may be easier to force video to the external display once macOS recovery mode is entered (still may need Clamshell mode activated once the recovery utilities screen is reached).

Unfortunately Apple does not make this easy like it is with non-Apple laptops which include a button to toggle the external display on & off.

Good luck.

One might be able to do this via a USB Installer of Big Sur and connected to External Display. Caution thought - Extra Special Notation regarding the Touch ID equipped Apple Computer. About Startup Security Utility and Must Enable from Recovery Mode the ability to boot from External Drive Before Attempting 

marc-antoine163 wrote:

Thank you. I created a Big Sur installer and I did everything I could but it's still not showing on the external display.

In order to boot from a USB drive you first need to boot into Recovery Mode and use the Startup Security Utility to allow booting from USB, otherwise your USB installer is worthless on a 2019 Mac.

About Startup Security Utility - Apple Support

Okay so right now the Mac no longer has a display. I removed it. When I turn it on, it automatically connects to the external display, but once the recovery has downloaded, and it's switching to the recovery mode, it says "No signal".

Try connecting to a different USB-C port. Perhaps even try rotating the USB-C connector of the display cable 180 degrees upside down in case you have a half bad USB-C port. If you disconnected the built-in display, then everything should go to the external display unless you have a USB-C port, cable, or monitor issue. Also make sure the built-in display cable did not partially reconnect.

I will most certainly repair the display myself, to save on labor cost and markup.

FYI, if you perform the display replacement yourself, then your Mac may not be able to work properly with the online video streaming services because a 2019 Mac with the T2 security chip requires the T2 chip to be programmed with the part serial numbers of all the major components. Perhaps this may even affect your Apple wallet and other "secure" Apple services. I really don't know what may be affected with a part not being associated with the T2 security chip.

In my experience most vendors selling Apple parts tend to price those parts near Apple's own prices so at most you are saving on labor fees.

Yes. If Apple suspects you have been inside the laptop they may decline to service the laptop (although that is probably illegal in most regions), but it won't stop Apple from giving you a hard time.

Find an Apple Authorized Service Provider nearest your location, Just fill in the Country and City Information.

Guess you followed the advise of @ HWTech ?

It may require the repair at this point !

Try disconnecting external Display and reconnect ??