What is the best time of the cycle (please be specific) to perform laparoscopy to cauterize endometriosis?

Early. The best time to perform the surgery is in the first 10 days of your cycle. If you are on birth control pills at the time, then it can be done later in the cycle as well.
Varies. Some physicians may prefer early vs late cycle and others may be the opposite. There is no particular best time in regard to effectiveness of the surgery - it really comes down to when it is most convenient for you.