As a note, you want the potatoes, carrots and celery to be roughly the same size. Keep this in mind when chopping the vegetables and deciding if you should quarter or halve the potatoes.
Season the chicken all over with salt and pepper. Season it with the smoked paprika and garlic powder.
Melt the butter with the olive oil in a large stock pot over medium heat. Add the chicken and cook until browned on both sides - about 5 minutes per side. Transfer the chicken to a plate.
In the same pot, add in the onions, garlic, carrots, mushrooms and celery. Add another pinch of salt and pepper. Cook, stirring often, until the veggies soften, about 6 to 8 minutes.
Pour in the white wine to deglaze the pan, stirring well to remove any brown bits on the bottom of the pot. Cook for 2 to 3 minutes.
Add the potatoes, put the chicken thighs back in and cover with chicken stock. Add a few springs of fresh thyme. Bring the mixture to a boil, then reduce it to a simmer and cover. Simmer for 30 minutes, covered, stirring occasionally.
After 30 minutes, use kitchen tongs or forks to shred the chicken in the soup.
Combine the milk and flour in a shake cup or jar, shaking for 30 seconds to combine. Stream it into the soup while stirring. Bring the mixture back to a boil for 5 to 10 minutes.
Taste and season with more salt and pepper if needed. Remember, the soup is only as flavorful as your stock, so you may need to add quite a bit more salt!
Serve immediately with chopped fresh parsley.