The Halal status of chicken and meat may be verified by knowing the producer. If the producer is certified by a valid Islamic organization and their Zabiha meets the conditions of Islamic slaughter then it would be considered Halal. If the Zabiha does not meet the Islamic condition of slaughter then it would not be considered Halal. If the chicken or meat is produced by an individual then one would have to make sure that the slaughterer was a proper Muslim who performed the Zabiha by invoking the name of Allah at the time of slaughter.
If you inquire from the restaurant, mostly they will share with you the source of their chicken and meat.
The oil, if it is used to fry or cook a non-Zabiha meat or chicken then it would become Najis (impure) because non-Zabiha meat and chicken is not only Haram but it is also considered Najis. If the Halal chicken or meat was fried and cooked in the same oil wherein the non-Zabiha chicken or meat was fried then the Halal chicken would also become Najis and impure. Thus it would be impermissible to eat from it.
The oil being used to fry anything has to be Halal and pure. All vegetable oil, canola oil, olive oil, corn oil etc.... is Halal. If the source of oil is the fat of an animal then the animal has to be Halal and it has to be slaughtered Islamically.
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America