Heat oil in large deep pot over medium-high heat.
Add carrots, celery, and onion; sauté until golden and tender, about 10 minutes.
Add garlic, thyme, bay leaves, and peppercorns; stir 2 minutes.
Add 8 cups water, 1/2 cup coarse salt, and sugar; bring to simmer, stirring until salt and sugar dissolve.
Add remaining 8 cups water and 4 cups ice cubes.
Let stand until brine is cooled completely.
Submerge chicken in brine; cover and chill at least 12 hours and up to 24 hours.
Remove chicken from brine.
Rinse chicken and pat dry; place in roasting pan.
Sprinkle chicken with pepper; roast until cooked through and juices run clear when thickest part of thigh is pierced, about 1 hour 20 minutes.
Transfer chicken to platter; cool slightly.
Spoon fat from roasting pan into small container.
Cover; freeze fat until firm; reserve for puff pastry.
Discard skin and bones.
Tear chicken meat into large bite-size pieces; reserve for filling.
DO AHEAD Chicken fat and chicken pieces can be made 1 day ahead. Keep fat frozen.
Cover and chill chicken.
Combine milk and garlic in small saucepan; simmer 2 minutes.
Remove garlic from milk and chop finely; discard milk.
Whisk 1/3 cup water and cornstarch in small bowl to blend; set aside.
Heat 1 tablespoon oil in heavy large pot over medium-high heat.
Add mushrooms and sauté until tender and brown, about 6 minutes. Transfer mushrooms to bowl.
Add remaining 2 tablespoons oil to same pot; add carrots, leeks, and potatoes and sauté until beginning to soften, about 10 minutes.
Add chopped garlic and thyme; stir 2 minutes.
Add 3 1/2 cups broth; bring to boil.
Reduce heat, cover, and simmer until vegetables are tender, about 15 minutes longer.
Rewhisk cornstarch mixture and add to vegetable mixture; stir until broth boils and thickens, about 2 minutes.
Stir in chicken, mushrooms, peas, and cream; simmer uncovered until heated through and mixture has consistency of stew, adding more broth by 1/4 cupfuls if mixture is too thick, 6 to 7 minutes.
Stir in lemon juice and hot pepper sauce.
Season to taste with salt and pepper.
Mix in truffle oil, if desired.
Divide pot pie filling among six 2-cup ramekins or soufflé dishes.
DO AHEAD Can be made 6 hours ahead.
Top each pot pie with 1 baked puff pastry round.
Bake pot pies until filling is heated through, about 15 minutes if warm or at room temperature and 25 minutes if chilled.