For the chicken: Preheat the oven to 350 degrees F.
Sprinkle the chicken breasts with salt. In a large saute pan, heat enough clarified butter to coat the pan over medium-high heat. Add half the thyme and continue to heat. Place 2 of the chicken breasts in the pan skin-side down and sear until golden brown, basting the flesh side with fat from the pan. Flip the breasts, brown on the other side and baste. Transfer the chicken to a baking sheet. Cook the other 2 breasts in the same manner, adding in the remaining thyme sprigs.
Place the baking sheet with the seared breasts in the oven and bake until the chicken is firm to the touch, 7 to 8 minutes. Allow to rest before slicing.
For the just: While the chicken is in the oven, pour off any excess fat in the saute pan and place over medium-high heat. Add enough clarified butter to coat the bottom of the pan and add the bacon lardons. Cook until crispy, and then remove to a plate and reserve. Remove any excess fat from the pan if necessary. Add in the onions and sweat until soft and translucent, about 5 minutes. Increase the heat, add in the mushrooms and season with salt. Cook the mushrooms until golden brown on all sides, 3 to 5 minutes.
Add in the white wine and scrap up all the brown bits that have collected on the bottom of the pan. Cook down until the wine has reduced by half. Add in half of the chicken stock and let simmer about 5 minutes. Add in the remaining chicken stock and reduce until slightly thickened Remove from the heat and swirl in the butter. Taste and season if necessary.
Serve the sliced chicken with the jus and garnish with chopped parsley.