Cook the potatoes in salted boiling water (your water should be "salted like the sea") until they become really soft and practically fall apart when you poke them with a fork.
While the potatoes are cooking, heat the olive oil in a saute pan set over medium heat and then throw in the onion, carrots, celery and garlic; cook until fragrant and slightly softened, about 3 minutes.
Tear the cooked chicken into strips with your fingers and add it to the pan, along with the chicken broth and water; bring to a low boil, cover, lower the heat and simmer slowly for about 8 to 10 minutes, or until the liquid is practically all gone. Stir in the frozen peas and corn kernels and set aside.
When the potatoes are cooked, drain them well and return them to the saucepan. Whisk the milk and sour cream together in a small measuring cup and pour about half of this mixture over the potatoes.
Beat with an electric mixer on low speed until the potatoes break down and the puree starts to come together, then increase the speed to medium. Add the rest of the milk mixture, a few tablespoons at a time, mixing well between each addition, until the puree becomes nice and fluffy. Adjust seasoning as needed by adding salt and white pepper, to taste.
Spread the mashed potatoes over the reserved chicken mixture; if you want to get real fancy about it, you can pipe them with a pastry bag equipped with a large open star tip)
Bake in a 375°F oven for 30 minutes, then broil for 3-5 minutes, or until the top becomes nice and golden (you can drizzle a little bit of olive oil over the potatoes to help them brown better).
Remove from the oven and allow to rest for 5 to 10 minutes before serving.