In what country did the pysanky egg art begin?

"Pysanky" is a Ukrainian term, so technically the answer to your question is that pysanky egg art began in Ukraine.
Decorated eggs have traditionally been created by Slavic people of the many nations and ethnic groups of what is now Central and Eastern Europe. The wax resist and dye method reached its greatest perfection in the territory now known as Ukraine. Each region within Ukraine has its own particular style and traditional colors, symbols, and designs. The intricate and detailed designs that most of us associate with pysanky developed from styles common to the Hutsul people of the Carpathian mountain region of Ukraine.

Incidentally, some ethnographers postulate that the original homeland of the Slavic peoples was in the territory now occupied by Ukraine, most likely Western or Northwestern Ukraine.