Heat a large heavy-bottomed pot over medium heat and add a splash of oil and a knob of butter, letting them melt together until they foam and smell nutty, about 2 minutes.
Add the sliced mushrooms to the pot and cook, stirring occasionally, until they release their juices and turn a deep golden brown, about 8-10 minutes. They should smell earthy and look caramelized.
Push the mushrooms to the side of the pot and add the chicken thighs, skin-side down if applicable. Cook until browned on one side, about 4-5 minutes, then flip and cook for another 4-5 minutes until just cooked through. The chicken should be tender and slightly browned.
Add the minced garlic and chopped thyme to the pot, stirring constantly for about 1 minute until fragrant and slightly softened.
Pour in the broth, scraping up any browned bits from the bottom of the pot to develop flavor. Bring to a gentle simmer and cook for 10 minutes, allowing the flavors to meld and the chicken to finish cooking.
Remove the chicken thighs from the pot, let them rest briefly, then shred or chop into bite-sized pieces. Return the chicken to the soup.
Stir in the heavy cream slowly, ensuring the soup stays at a gentle simmer to prevent curdling. Mix until the broth becomes velvety and rich.
Taste and season with salt and pepper as desired. Let the soup simmer for another 2-3 minutes to meld all flavors together.
Serve the soup hot, garnished with additional thyme or fresh herbs if you like. It should be creamy, with tender chicken, deeply browned mushrooms, and a fragrant, silky broth.
