I reside in USA. A few days back I went to a Christian farmer who has a poultry farm. There I saw how they slaughter chickens. They first give it an electric shock so it becomes paralyzed. After that they behead it with a single stroke of heavy knife and then they hang it for a while to let the blood drain off. I asked the farm-owner that do you say name of God while slaughtering the chicken so he said, "yes, we do". But I was noticing the guys who were slaughtering chickens that they were doing everything too quickly and I doubt if they were saying the name of God. So after knowing all these details, is it still permissible to eat chicken meat from that particular farm? If chicken meat from this farm is Haram then what is the ruling regarding meat from other farms as now there is a doubt that most probably other farms will be doing the same method?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The questioner mentioned three doubts in his question.
1) Giving an electric shock to paralyze chickens to make it easy to slaughter them.
In this case, if chickens die due to the electric shock the ruling of killing in this manner is that eating such meat is forbidden in Islamic Shari'a.
If the electric shock only paralyzes the chickens and does not kill them, then eating them is permissible if they are slaughtered Islamicly after the shock.
2) Beheading chickens by a single stroke of a heavy knife. According to Sunnah the animals should be slaughtered with a sharp knife. Therefore, doing so is better and desirable. However, slaughtering by a sharp, heavy knife is permissible provided the act of killing occurs by the sharpness of the knife not by its weight. Slaughtering by any sharp means is permissible whether it is iron, wood, and stone, etc. with the exception of nails and bones. Cutting off the heads of chickens, animals while slaughtering them is permissible though some scholars dislike it if it is done intentionally.
3) Mentioning the name of Allah at the time of slaughtering or uncertainty of mentioning it.
Supposed you should not have asked them whether they mentioned the Name of Allah at the time of slaughtering or not. Their confession of slaughtering is sufficient as Imam Nawawi said: 'If a dissolute Muslim or a person from the People of the Book admits that he has slaughtered the animal, we should accept his confession and eating such meat is permissible since he is among those whose slaughtered animal is Halal' [al-Majmu'a]. As one can note that Imam Nawawi did not stipulate the condition of mentioning Allah's Name as long as he admits of slaughtering.
However, the matter of mentioning name is not so serious since the Prophet Muhammad's (Sallallahu Alaihi wa Sallam) manners were to accept the food and meat presented by the People of the Book without making any inquiry whether they mentioned the Name of Allah or not?
If we begin to question about every animal we will exert pressure on us while Allah does not place burdens on us.
Imam Zuhri said concerning the matter of slaughter carried out by the People of Book: 'If you hear mentioning of other than Allah's Name do not eat it; otherwise, Allah made it Halal for you in spite of their Kufr'. (See: Fath al-Bari, Vol.: 9, page: 137).
In brief, one should confirm only two matters:
A) The slaughterer is a Muslim or from the People of the Book.
B) The electric shock should not cause the death of chickens before slaughtering them.
Allah knows best.