Which feast did Josiah keep as none had been kept since the days of the judges?

Answer: Passover

Josiah commanded the Passover as written in the Book of the Covenant; no such Passover had been kept since the days of the judges.

“The king commanded all the people, “Keep the Passover of the LORD your God, as it is written in this Book of the Covenant.” No such Passover had been observed from the days of the judges who had governed Israel through all the days of the kings of Israel and Judah.”— 2 Kings 23:21-22 (BSB)