In a large deep skillet or Dutch oven, melt the butter over medium heat.
Add the sliced mushrooms and cook for 5 minutes, stirring occasionally, until they release their moisture and begin to brown.
Add the minced garlic and cook for 1 minute until fragrant.
Stir in the cooked chicken and broken spaghetti, mixing to combine evenly.
Pour in the chicken broth, whole milk, and heavy cream. Stir to combine all ingredients.
Add the salt, black pepper, dried thyme, and crushed red pepper flakes. Stir well.
Bring the mixture to a gentle boil, then reduce heat to low. Cover and simmer for 15-18 minutes, stirring occasionally, until the spaghetti is tender and most of the liquid is absorbed.
Remove the lid and stir in the grated Parmesan cheese until melted and the sauce is creamy.
Taste and adjust seasoning if needed.
Sprinkle chopped fresh parsley over the top before serving.