It’s incredibly simple. You just have to visit any office of the health insurance company you chose to register with (in my case, Dôvera) and apply for one. It takes about one month to be issued and you can choose to have it sent home by post or to collect it in person. I went to collect it myself. They asked for my paper health insurance card and gave me a proper plastic card instead. My new card is valid for five years and has the European Health Insurance (Európsky preukaz zdravotného poistenia) information on one side and the Dôvera Health Insurance information on the other side.
What if you need it earlier than that? Worry not, they will print out a paper for you, as an alternative to the EHIC, valid for three months. You can use that paper to access public sector healthcare services during temporary visits abroad, within the EU.
Simple as that 😉
I assume everybody here knows what the EHIC is.