Fill Dutch oven or large frying pan halfway with oil and butter; heat over medium heat.
In large bowl, mix eggs and milk then place chicken strips in mixture.
In separate bowl, mix flour, barbecue seasoning and mesquite seasoning. Add salt and pepper, to taste.
Remove 4-6 chicken strips from milk mixture and cover completely in flour mixture.
Drop pinch of flour in hot oil; if it crackles, it’s hot enough. Once hot, fry battered chicken 4 minutes then, using tongs or fork, turn over and fry 4 minutes. When done, chicken will be golden and almost float. Remove from frying pan and repeat with remaining chicken.
Serve with dipping sauce of choice, if desired.