You have to beat the first boss without draining their health, but instead filling up the green bar underneath the healthbar.
The Healthbar of the boss at the end of the fight should look something like this:
There needs to be a bit of health left but the blue bar needs to be filled, you can achive that by beating the boss as fast as you can.
After that another cutscene should play that starts like this:
For video help, here is a lets play where someone achived it (around 4:50 in the video):
hope this helps!