In a nonstick pan, melt 1 tbsp each of butter and olive oil. Add the shredded chicken, sprinkle with salt and pepper, and cook until you start to get some color to the chicken, about 5 minutes. Remove the chicken to a large bowl.
In the same pan, melt another tbsp each of butter and olive oil. Add the mushrooms, sprinkle with salt and pepper, and saute until brown, about 10 minutes. Add the onion, garlic, and thyme, and saute until the onions are translucent, about 5 minutes, adding some more olive oil if needed. Add the wine, turn the heat to high, and cook until the wine is evaporated, about another 5 minutes. Remove from heat and add to the same bowl as the chicken.
Meanwhile, preheat the oven to 425° F and bring a large pot of salted water to a boil and cook the linguine until it's almost done, about 9 minutes (it'll finish cooking in the oven).
Next add 3 tbsp butter, 1 tbsp olive oil, and 4 tbsp flour to the pan over medium heat. With a silicon whisk, cook the roux until golden brown, about 5 minutes. Add the milk, heavy cream, chicken stock, 1 tsp salt, 1 tsp pepper, and the nutmeg to the pan. Cover and turn the heat to high and bring to a boil. Uncover and simmer until thickened, about 5 minutes. Add the peas and parsley.
Add the sauce to the bowl with the chicken and mushrooms, stir to combine. Spray a 9x13 pan with canola oil, and add the mixture. Top with the breadcrumbs and parmigiano reggiano. Cook until golden brown, about 25 minutes. Remove from the oven and let sit at least 10 minutes before serving.