Preheat oven to 300°F and make enough room to accomodate a Dutch oven with lid.
Pat the chicken thighs dry and season with salt and pepper on all sides. Heat a large Dutch oven to medium heat. After the pot is hot add 4 tablespoons of olive oil and once shimmering, add the chicken thighs. Sear the chicken thighs for around 5-7 minutes per side or until well browned. Work in batches and do not crowd the pot.
Set the seared chicken thighs aside and turn the heat down to medium-low. Add in the carrot, celery, and onion and saute until very soft (about 10-15 minutes) then add the garlic and cook for 2 minutes longer.
Next, add in the tomato paste and cook for 5 minutes, stirring occasionally. If the paste starts to burn, add a splash of water. Add the wine and turn the heat to high. Using a wooden spoon scrape all of the brown bits off the bottom of the pot.
After cooking the wine for 3 minutes, add the plum tomatoes, thyme, rosemary, and parmesan rind and bring to a simmer. Once simmering add the chicken thighs to the pot, stir, then turn off and cover. Add the pot to the oven and braise for 45 minutes. After 45 minutes, remove the lid and continue to cook for 30 minutes.
Remove the pot from the oven. Using forks, shred the chicken thighs into bite sized pieces. Cover and set aside and cook the mushrooms.
Heat a large pan to medium high heat and add the mushrooms. Once the mushrooms release their water, add remaining 2 tablespoons of olive oil and saute until the mushrooms are brown. Season the mushrooms with salt and pepper then add them to the ragu. Taste test the ragu and make any final adjustments to salt and pepper if required.
Bring a large pot of salted water to boil and cook the pasta until 1 minute less than al dente. Heat a large pan to medium heat and add in 4 cups of the chicken ragu. Add the pasta to the pan and cook for 1 more minute mixing well to thoroughly coat the pasta.
Remove the pan from the heat and mix in the grated parmesan cheese. If the pasta needs more ragu, add more now. If the sauce seems a little dry add a bit of reserved pasta water to loosen it up. Serve plated with grated Parmigiano Reggiano and the remaining ragu on the side. Enjoy!