The Verdict: Clear Path
Apply for the long-term residence permit for scientific research, based on a hosting agreement with your Czech institution. Ask the institution to draw up the agreement and to put you through the Key and Research Staff Programme.
The research permit rests on the hosting agreement with an accredited research organisation, so it skips the vacancy registration and labour-market steps of the employee card. The Key and Research Staff Programme, run by the Ministry of Industry and Trade, does not change the test but guarantees consular capacity and simplifies documents; the institution applies to join it, not you.
If you will be employed on ordinary terms rather than hosted, the blue card is the alternative: higher education and a salary of at least 1.5 times the average gross annual wage announced by the Ministry of Labour each 1 May.
From either, permanent residence comes after five years and citizenship five years later, with the B1 Czech exam and the Czech Life and Institutions test. If your standing is exceptional, also ask the embassy about permanent residence in the interest of the Czech Republic; it has no published criteria, but would skip those first five years.