Combine raw chicken thighs and carrots in a pot, cover with water, and bring to a boil, reduce heat to low, and simmer for 15 minutes. Drain, separate the carrots and chicken and shred the chicken.
Add the potatoes to a pot with salted water, the water should have enough salt so it tastes like the sea. Bring to a boil until the potatoes are easily pierced with the tip of a knife. Drain, add the butter, half & half, salt, and pepper. Mash until smooth and creamy.
While potatoes are boiling, begin to saute the mushrooms, onions, and celery in butter and olive oil, in a large heavy-bottomed skillet over medium-high heat. Saute until the mushrooms have released all of the liquid and begin to brown.
Add the flour to the onion mixture and stir to completely coat all the onions. Slowly add the chicken stock gently stirring to make a smooth sauce.
Remove the skillet from the heat and add all the ingredients. Stir to combine.
Place the filling in an oven-proof casserole or skillet. A 12-inch castiron skillet is a perfect size.
Add spoonfuls of mashed potatoes around the top of the filling and gently smooth it over the surface with the prongs of a fork to give it a nice texture.
Place Shepherd's pie on a sheet tray to catch and drips, in a preheated 350°F oven for 30 minutes. Until the topping has some color.
If you want more color on the topping you can place it under the broiler. Keep a close eye on it so it doesn't burn.
Serve with a sprinkling of chopped parsley