For the pesto: Place hazelnuts on sheet tray and toast in the oven for 6 to 8 minutes or until golden. Let cool. Place hazelnuts, garlic, and basil in the bowl of a food processor. With the motor on, add oil in a steady stream and process until smooth. Season to taste with salt. Set aside.
For the lamb chops: Season chops generously with salt and pepper. Add the oil to a cast-iron pan set over medium heat. When oil is very warm, but not too hot, add the butter; when foaming, add the chops flat-side down. Cook until chops are brown and crispy, about 3 minutes per side.
To serve, top each chop with approximately 1 tablespoon of the pesto.