In a large heavy skillet, heat the olive oil. Add the shallots and cook over moderately high heat, stirring, until browned, 8 to 10 minutes; transfer to a plate.
Melt the butter in the skillet. Season the chicken pieces with salt and pepper. Add them to the skillet skin side down and cook, turning once, until browned, about 12 minutes; transfer the chicken to a plate. Discard the fat.
Return the chicken to the skillet with the shallots, stock and thyme. Cover and cook over moderately low heat for 20 minutes. Transfer the wings, drumsticks and any soft shallots to a platter. Cover and cook the rest of the sauté for another 5 minutes. Transfer the remaining chicken and shallots to the platter. Skim the fat from the pan juices, season with salt and pepper and pour over the chicken. Sprinkle with the parsley, garnish with the lemon wedges and serve.
Buckwheat groats, also known as kasha. Use some of the rendered chicken fat to sauté the kasha before boiling it. Or serve couscous.
Pair with a citrusy California Sauvignon Blanc, like Matanzas Creek Sauvignon Blanc.