I'm assuming this is an Intel Mac since you are using Command + Option + R.
Try connecting an external wired USB Keyboard & Mouse. Power on the laptop & close the clamshell lid while also pressing & holding the Option key. Timing can be tricky. Usually I like to do all this immediately after hearing the startup chime. If you see the Option Boot screen (aka Apple boot picker menu) on the external display, then press Command + Option + R.
Trying to boot with a damaged built-in Display can be very tricky & difficult. Sometimes it can even be impossible, but with a 2017 or older Mac it should be possible. A 2018-2020 Intel Mac.....may have even more trouble with a broken screen, but usually it should usually be possible to get video on the external..
Keep in mind that the system may have actually booted into Internet Recovery Mode without you realizing it because the built-in Display may have gotten stuck on the Apple logo & is no longer being update. I've seen this happen a few times. If you allow it enough time to boot, then you can try pressing Command + F1, or Command + Brightness Down).....you may need to involve the Fn key as well depending whether you have a Touchbar model.