In a pot of boiling, salted water, blanch the green beans for 1 minute. Remove and plunge into iced water. Drain well and reserve.
In a saucepan on a low heat, warm 1 tbsp olive oil and cook eschalots with salt until softened (about 8 minutes). Stir through blanched beans, then remove saucepan from heat. Reserve.
Warm remaining oil in a frying pan over a medium heat. Pan-fry the kingfish fillets for about 2 minutes each side (cooking time will depend on the thickness of the fillets). Remove fillets from heat and allow to rest somewhere warm.
Place saucepan with beans back on a medium heat; warm through for a minute.
Place butter in a separate frying pan on a medium heat. Once it begins to foam and just as it is starting to turn brown (about 5 minutes), add vinegar, hazelnuts and a pinch of salt. Allow to cool.
Place kingfish fillets on four plates, top with beans and eschalots, and spoon butter sauce and hazelnuts over the fish. Serve immediately with freshly ground pepper.