Are there any Hadeeths about the virtue of fasting the first ten days of Muharram? May Allah reward you.
All perfect praise be to Allah, the Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.
It was narrated on the authority of Abu Hurayrah, may Allah be pleased with him, that the Prophet said: "The best fasting after Ramadan is the month of Muharram." [Muslim, An-Nasaa’i, Abu Daawood and others] An-Nawawi said that this Hadeeth is a direct indication that Muharram is the best month for fasting (after Ramadan).
The best days of the month of Muharram are the first ten days. Al-Mardaawi said in Al-Insaaf, "The best days of Al-Muharram are the tenth then the ninth then the first ten days of it.”
Ibn Rajab said in Lataa’if Al-Ma‘aarif, "The best days of Muharram are the first ten days." He quoted Abu ‘Uthmaan An-Nahdi as saying, “They (the people) would glorify three tens: the last ten days of Ramadan, the first ten days of Thul-Hijjah and the first ten days of Muharram”.
However, there is not any authentic Hadeeth regarding the virtue of fasting the first ten days of Muharram altogether.
Allah Knows best.