Combine the chicken and stock together in a large pot and simmer until the chicken is tender, 1-1½ hours.
Remove the chicken from the stock and cool. Add the onions, carrots and celery to the stock and simmer 7-10 minutes. Remove the vegetables from stock and set aside. Cook the peas or asparagus and mushrooms in the stock for 2 minutes; remove the vegetables and set aside. Reserve the stock.
When the chicken has cooled, remove the meat from the bones. Discard the bones and skin and reserve the meat.
Melt the butter in a saucepan, stir in the flour and cook gently for 2-3 minutes, stirring. Whisk in 4-6 cups of the stock along with the milk and continue cooking over medium heat, stirring, for 3-5 minutes until it lightly coats the back of a spoon. If it is too thick, add more stock. Season to taste with salt and pepper. Set aside.
Heat the olive oil in a large, heavy nonstick skillet. Fry the sausages whole over medium heat for 5 minutes, turning often, until firm and browned on all sides. Cut into ¾ inch rounds.
Preheat the oven to 425°.
To make the biscuit dough, combine the flour, baking powder, salt and sugar in a medium-sized mixing bowl. Cut in the butter until the mixture resembles coarse crumbs. Pour in the half-and-half and blend gently. Knead the soft dough four or five times on a floured surface to form a ball and sprinkle lightly with flour.
To assemble the pie, roll the crust ½ inch thick and large enough to cover the surface of a deep baking dish. Arrange the cooked chicken, sliced sausage and vegetables in the baking dish. Pour in the gravy, cover with the crust.