Final answer:
The Judge who made a rash vow to the Lord which led to offering his own daughter as a burnt offering was Jephthah.
Step-by-step explanation:
The Judge who made a rash vow to the Lord saying, "whatever" came out of his house to greet him upon his return from battle "will belong to the Lord, and I will offer it as a burnt offering" is Jephthah. This event is mentioned in the biblical book of Judges, specifically in JG 11:29-39. Jephthah's vow resulted in a tragic conclusion when his daughter, being the first to come out of his house to greet him, became the subject of his vow. This story highlights the serious consequences of rash promises, especially when they are made to God.