Season the chicken thighs liberally on all sides with salt and pepper. Rub the chicken with 1 tablespoon of the oil. Roast in the oven, skin-side up, until the chicken is cooked through, about 30 minutes.
Meanwhile, to the bowl of a food processor, add the remaining 3 tablespoons olive oil, pistachios, Parmigiano-Reggiano and lemon juice. Pulse until the pistachios are finely chopped and the pesto begins to stick together. Serve alongside the roasted chicken thighs.
The salt-and-pepper-seasoned chicken can be kept in the refrigerator uncovered for up to two days. Add the oil before roasting.
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