It is a pretty standard French chicken sauté dish really. The thing that makes it Basque is the addition of Piment d’Espelette, a Basque type of chilli that is not that hot but is essential in the cooking of the region.
There are lots of recipes for this, mine uses onions, garlic, red and yellow peppers, a decent splash of white wine, chopped tomatoes (plus some tomato puree if you have pallid tomatoes), chicken pieces (legs are good but we had breasts tonight), the piment and some dried herbs (I like herbes de Provence).
It is a pretty simple dish, you sauté the chicken in olive oil and then cook the vegetables down in the same oil and put the chicken back and simmer the dish covered until the chicken is done.
I like to add in some olives too, towards the end of the cooking time.
We ate this with plain rice and drank a nice rich Côtes du Rhône with it, but any southern French red would be excellent.