Preheat oven to 375°F Heat 2 Tbsp of olive oil in a large sauté pan on medium high or high heat (hot enough to brown but not burn). Sprinkle a dash of salt on the bottom of the pan. Season the chicken pieces all over with salt and pepper. Working in batches, brown the chicken pieces on two sides, about 1-2 minutes per batch. Add a little more salt to the pan (and more olive oil if needed) after every batch. This will help prevent the chicken from sticking to the pan. Remove chicken pieces and set aside in a bowl. Note that the chicken does not have to be cooked through, only browned.
In the same sauté pan add 1 Tbsp olive oil, lower the heat to medium, add the onions, and cook until translucent, about 3 minutes. Add the garlic, cook 30 seconds more. Remove onions and garlic to a shallow (9 x 13 x 2) casserole dish.
Raise heat to medium high, add the sliced mushrooms. Dry sauté them (no need to add butter or oil), allowing the mushrooms to brown lightly, and release some of their moisture. Add the mushrooms to the casserole dish.
Add 1/4 cup dry sherry or dry white wine to the pan to deglaze the pan, scraping off the browned bits from the bottom of the pan. (At this point, if you are making ahead, reduce the sherry to 1 Tbsp and pour off into the casserole dish. Refrigerate cooked onions, garlic, mushrooms, and browned chicken pieces until you are ready to make the casserole.) Let the sherry reduce to about 1 Tbsp, then add the chicken stock, and remove from heat. Stir in 1 1/2 teaspoon salt, the cream, and the sour cream. Add the raw rice to the casserole dish. Then pour the stock, sherry, cream, sour cream mixture over the rice. Add the Italian and poultry seasonings (or fresh herbs) and paprika to the dish. Stir the rice, onion, mushroom, herb mixture so that they are evenly distributed in the casserole dish.
Place the chicken pieces on top of the rice mixture (in a single layer if you can, they will be crowded). Cover the casserole dish tightly with aluminum foil. Bake in a 375°F oven for 45 minutes. Remove foil. If the casserole is still too liquidy, let it cook a few minutes more, uncovered, until the excess liquid has evaporated away.