Heat the oven: Preheat oven to 375°F (190°C).
Season the chicken: Pat the 4 bone-in skin-on chicken thighs (about 2 lbs total) dry and season generously all over with 1 teaspoons kosher salt, plus more to taste and 0.5 teaspoons freshly ground black pepper, plus more to taste.
Sear the chicken: Heat a large oven-safe skillet over high heat. Add 1 tablespoons olive oil and swirl to coat. Add the chicken skin side down and sear, flipping once, until deeply golden on both sides, about 8 minutes per side.
Remove chicken to a plate. Drain all but 1 tablespoon of fat and lower the heat to medium-high. Add 1 tablespoon of the 2 tablespoons unsalted butter, divided, then the 1 large sweet onion, coarsely chopped, 3 large shallots, coarsely chopped, 2 tablespoons finely chopped ginger, 4 garlic cloves, skin-on, crushed, and 1 serrano chile, finely sliced (or 1/2 fresno chile). Season with salt and pepper and sauté, stirring, until softened, about 5 minutes
Make a well in the center of the pan and add the remaining 1 tablespoon 2 tablespoons unsalted butter, divided. When melted, add the 2 teaspoons garam masala, 2 teaspoons madras curry powder, 0.5 teaspoons ground cinnamon, 0.3 teaspoons ground turmeric, 2 tablespoons tomato paste, and the finely chopped stems from the 1 cilantro, stems finely chopped and leaves coarsely chopped. Toss to coat with butter and cook until fragrant and toasted, about 30 seconds
Add the 2 cups low-sodium chicken broth, scraping the bottom of the pan to deglaze, and bring to a boil. Return the chicken to the pan skin side up, then transfer to the oven. Cook until the sauce is reduced and the chicken is falling off the bone, about 50 minutes
Return the pan to the stovetop and divide the chicken among serving dishes. Set the pan over medium heat, add the 0.3 cups whole milk Greek yogurt (or sour cream), and whisk to incorporate. Season the sauce with salt and pepper and stir in the coarsely chopped 1 cilantro, stems finely chopped and leaves coarsely chopped leaves, reserving 1/4 cup. Spoon the sauce around the chicken, top with the reserved cilantro, and serve with 4 cups cauliflower rice, for serving.