Pat the chicken thighs dry with paper towels, then season generously with salt and pepper. Heat the large skillet over medium heat and add 1 tablespoon of butter. Once melted and bubbling, place the chicken thighs in the pan, skin side down if applicable, and cook until golden brown on each side, about 5-7 minutes per side. Remove and set aside.
While the chicken cooks, trim and thinly slice the leeks, then rinse thoroughly in cold water to remove dirt. Add the remaining 1 tablespoon of butter to the skillet and stir in the leeks. Sauté until they become soft and translucent, about 5 minutes, then add the minced garlic and thyme, cooking for another minute until fragrant.
Pour in the chicken broth, scraping up any browned bits from the bottom of the skillet with a wooden spoon. Bring the mixture to a gentle simmer and return the browned chicken thighs to the skillet, skin side up. Cover and reduce heat to low. Simmer for 30 minutes, or until the chicken is cooked through and tender.
Once the chicken is cooked, remove it from the skillet and set aside. Increase the heat to medium, stirring in the milk or heavy cream, and simmer the sauce uncovered until it thickens slightly and becomes velvety, about 5-7 minutes. Taste and adjust seasoning with salt and pepper as needed.
Return the chicken to the skillet, spooning some of the sauce over the top. Continue cooking for another 5 minutes to reheat and allow flavors to meld. Serve hot, garnished with additional thyme if desired, with the sauce spooned over for a glossy, tender final presentation.
