Factory Pattern but with object Parameters

Take the following classic factory pattern:

public interface IPizza
{
    decimal Price { get; }
}

public class HamAndMushroomPizza : IPizza
{
    decimal IPizza.Price
    {
        get
        {
            return 8.5m;
        }
    }
}
public abstract class PizzaFactory
{
    public abstract IPizza CreatePizza(ItalianPizzaFactory.PizzaType pizzaType);
}

public class ItalianPizzaFactory : PizzaFactory
{
    public enum PizzaType
    {
        HamMushroom,
        Deluxe,
        Hawaiian
    }

    public override IPizza CreatePizza(PizzaType pizzaType)
    {
        switch (pizzaType)
        {
            case PizzaType.HamMushroom:
                return new HamAndMushroomPizza();
            case PizzaType.Hawaiian:
                return new HawaiianPizza();
            default:
                throw new ArgumentException("The pizza type " + pizzaType + " is not recognized.");
        }
    }
}

What if one (or many) of the Concrete Pizzas requires a parameter specific to the concrete implementation at construction. For example, lets say the HamAndMushroom factory requires a parameter called, MushroomType and this parameter would be required to instantiate the object?