Pat the chicken thighs dry with paper towels and generously season with salt & pepper. Heat the oil in a dutch oven or large pot over medium-high heat. When the oil is hot sear the chicken thighs skin-down down until skin is well browned, about 5 minutes. Transfer chicken to a plate and set aside.
Reduce the heat to medium and add the butter to the pot. When the butter has melted add the onion and mushrooms to the pot and cook, stirring occasionally, until browned and most of the liquid has evaporated from the pot, about 10 minutes.
Stir in the garlic and cook until fragrant, about 10 seconds. Add the flour and cook while stirring until well combined. Add the chicken stock, rosemary, thyme, bay leaf, and 1/2 tsp salt into the pot and bring to a boil over medium-high heat.
Return the chicken to the pot and reduce heat to medium-low. Cover the pot and simmer, stirring occasionally, until chicken is cooked through and very tender, 35 to 40 minutes.
Transfer the chicken thighs to a plate and shred the chicken using two forks discarding the skin and bones. Return the shredded chicken to the pot and stir to combine. Pour in the heavy cream while stirring and cook until soup is heated through, about 3 minutes. Season with salt & pepper to taste.