Season chicken thighs generously with salt and pepper.
Preheat a large skillet over high heat. Add olive oil and swirl to coat. Add chicken thighs, skin side down, and saute, flipping once, until deeply golden on both sides, about 4 minutes per side.
Remove to a plate, drain all but 1 tablespoon fat, and lower heat to medium-high. Add 1 tablespoon butter, onion, shallots, ginger, garlic, and chile. Season generously with salt and pepper and saute, stirring, until softened, 5 minutes.
Make a well in the center of the pot and add remaining 1 tablespoon butter. When melted, add garam masala, madras curry powder, cinnamon, turmeric, tomato paste, cilantro stems, and toss to coat with butter. Cook until fragrant and toasted, 30 seconds more.
Add chicken stock, scraping bottom of pan to deglaze, and bring to a boil. Return chicken to pan, skin side up, then transfer the pan to the oven. Cook until sauce is reduced and chicken is falling off the bone, about 50 minutes.
Return pan to stovetop and divide chicken among serving dishes. Place pan on medium heat. Add yogurt and whisk to incorporate. Season sauce with salt and pepper and add all but 1/4 cup cilantro. Spoon sauce around chicken, top with reserved 1/4 cup cilantro, and serve with cauliflower rice.