Heat the oven to 375°. Coat the chicken with the oil; arrange the pieces, skin-side up, in a large roasting pan. Sprinkle the chicken with the oregano, lemon juice, salt, and pepper. Top each piece of chicken with a piece of the butter.
Cook the chicken until the breasts are just done, about 30 minutes. Remove the breasts and continue to cook the legs until done, about 5 minutes longer. Remove the roasting pan from the oven; return the breasts to the pan. Top the chicken pieces with the feta cheese. Press any cheese that rolls off into the pan back onto the chicken. Baste the chicken with the pan juices.
Heat the broiler. Broil the chicken until golden brown, about 2 minutes. Serve with the pan juices.
Menu Suggestions: Balance the tanginess of lemon and feta with a mild side dish such as orzo tossed with a little olive oil or sautéed zucchini.
Wine Recommendation: This Greek-flavored dish will go nicely with a number of rustic, spicy red wines. Try finding a bottle from the Greek island of Paros or Santorini. Another alternative would be a syrah-based wine such as a Crozes-Hermitage from the northern Rhône Valley in France.