Preheat oven - Preheat oven to 350 degrees F. and spray a large casserole dish with nonstick cooking spray. Set aside.
Shred the turkey - gather approximately 5 cups of cooked turkey and shred into small, bite-size pieces (see notes).
Boil the spaghetti - Meanwhile, bring a large pot of salted water to a boil over high heat. Add the noodles and cook until just under al dente. Drain, rinse and set aside.
Sauté the vegetables - Heat the olive oil and 1 tablespoon butter in a large, heavy-bottomed Dutch oven over medium-high heat. Add the onion and cook for 2-3 minutes, or until starting to soften. Add the mushrooms and continue to cook for 3-4 minutes, or until tender. Stir in the minced garlic and cook for 1 minute. Add the white wine and bring to a simmer. Cook until liquid is reduced by half.
Prepare the roux - Over medium heat melt the remaining butter with the onions and mushrooms. Sprinkle with flour, whisking continuously to combine the flour with the melted butter, forming a thick paste covering the onions and mushrooms (this is your roux).
Make the sauce - Whisking continuously to prevent clumping, slowly add the chicken broth. Once all of the chicken or turkey stock has been incorporated, add in the heavy cream. Season with salt, black pepper, and fresh thyme. Bring to a low simmer then reduce heat to low.
Return the turkey - Add the turkey, frozen peas, and half of the cheese to the sauce and simmer all together for 2-3 minutes. Season with additional salt and pepper, to taste.
Assemble - Gently mix in the spaghetti noodles and mix before transferring everything to your prepared casserole baking dish. Sprinkle with additional thyme, if desired, and top with the remaining shredded cheddar cheese.
Bake - Transfer to the oven and tent with foil. Bake for 30 minutes covered, then remove the foil tent and continue to bake for an additional 15 minutes.
Serve - Allow the baking dish to rest for 5 minutes before serving. Enjoy!