I read about Tsar Nicholas II praising the Phanagoritsy regiment for shooting down some workers, but any research I do is not helpful at all in understanding what the Phanagoritsy regiment actually is, if someone could help or guide me to a useful source I would really appreciate it.
I assume your source for this is Trotsky? Phanagoritsy means "From Phanagoria". This almost certainly refers to the 11th Phanagoria Grenadiers, who Nicolas II reviewed several times, although I am unable to find any English language sources for them shooting workers. Apparently they were involved in fighting the Bolsheviks in 1917.
Source: The Unusual Award to the Phanagoria Grenadiers,Aleksandr Popov From Tseikhgauz No. 14, 2/2002. Pages 26-29, translated by https://www.marksrussianmilitaryhistory.info/PhanagoriaGren.htm