Preheat oven to 350°. Generously season quartered chicken with salt, pepper and paprika. Heat up an oven-safe skillet, add olive oil and place chicken skin down in the skillet. When the skin side is lightly browned, flip pieces over. When bottom side is browned, remove from stove and place into the pre-heated oven for 25 minutes.
Get a sauté pan nice and hot on the stove and add olive oil. Just as the olive oil is about smoke, add sliced mushrooms. The high heat will brown the mushrooms before they release their liquid, giving them good color and flavor. Once the mushrooms are seared on both sides remove them from the heat. Set mushrooms and liquid aside. Sauté the onions in the same pan until they are translucent. Just before they are done, add the garlic. Remove from heat and then add the mushrooms and juices back into the pan.
When the chicken is done remove from oven and place aside to rest. Deglaze the skillet with chicken stock. Add chicken stock to the onions and mushrooms and bring to a slow boil. Dissolve cornstarch into a quarter cup of cold water. Reduce heat to low, pour cornstarch mixture into pan and slowly stir until thickened.
Plate the chicken quarters and spoon mushroom gravy over the pieces. You can do this family-style or one plate at a time.