In a large bowl, combine the flour, paprika and 2 pinches each salt and pepper. Coat the chicken pieces with the seasoned flour, shaking off the excess.
In a large, deep skillet, melt the butter over medium heat. Add the chicken to the pan, skin side down. Cook, turning occasionally with tongs, until the skin is golden, 6 to 8 minutes. Transfer to a plate. Add the mushrooms and scallion whites to the skillet and cook, stirring often, until tender, 3 to 4 minutes. Stir in the chicken broth, scraping up the browned bits from the bottom of the pan, and bring to a boil. Return the chicken pieces to the skillet, cover and simmer over medium-low heat until the juices run clear, 20 to 25 minutes.
Stir the yogurt and scallion greens into the pan sauce; season with salt, pepper and paprika. Stir to coat the chicken with the sauce.
To serve, mound some rice on each of 4 plates and top with the chicken.