How to Make It
In a blender, puree the 1 1/2 tablespoons of cilantro, the chopped scallions, garlic, jalapeño, shallot, 1/2 cup of the fish sauce and 1/2 cup of the sugar until smooth. Season with the salt. Put the chicken in a large resealable plastic bag and pour in the marinade. Seal and refrigerate for at least 2 hours or up to 6 hours. Bring to room temperature 30 minutes before cooking.
Preheat the oven to 400° and line a large rimmed baking sheet with foil. In a large skillet, melt 1 tablespoon of the butter in 1 tablespoon of the oil. Add half of the chicken, skin side down, and cook over moderately high heat until the skin is brown and crispy, about 3 minutes. Reduce the heat to moderate if the chicken starts to brown too quickly. Transfer the chicken to the prepared baking sheet skin side up and repeat with the remaining butter, oil and chicken. Transfer the baking sheet to the oven and roast the chicken for about 25 minutes, until golden brown and cooked through. Let rest for 5 minutes.
Meanwhile, in a large bowl, whisk the lime juice with the Thai chile and the remaining 1/2 cup of fish sauce and 2 tablespoons of sugar. Add the papaya, carrot and tomatoes and toss to coat.
Transfer the papaya salad to a large platter or plates and top with the roast chicken thighs. Garnish with chopped cilantro, sliced scallions and peanuts; serve immediately.