Grease a 2-quart casserole and set aside.
In a large pot, melt the butter and heat the oil over medium-high heat. Add the onions, carrots and celery and cook, stirring, until soft, 3 to 4 minutes. Add the garlic, Essence, salt, and pepper and cook, stirring, for 30 seconds. Add the mushrooms and thyme and cook, stirring, until the mushrooms are soft and have given off their liquid, about 3 minutes. Add the sherry and cook until most of the liquid is evaporated. Stir in the flour and cook, stirring, for 2 minutes. Stirring constantly, slowly add the chicken stock and half-and-half and cook until the mixture is smooth and thickened, about 5 minutes. Add the chicken, peas, and parsley, stir well, and cook until chicken is heated through.
Fill a deep heavy pot halfway with oil and heat until oil is 360 degrees F. Shred the potatoes on a mandoline to make shoestring fries and fry until golden brown and cooked through. Remove from oil with a spider or slotted spoon and drain on a paper towel-lined plate.
Pour chicken mixture into a casserole dish and top with the shoestring potatoes. Serve hot.
