Pat the chicken thighs dry, then heat olive oil in a large Dutch oven over medium-high heat. Place the chicken skin-side down and sear for about 5 minutes until golden brown, then flip and sear the other side for another 5 minutes. Remove the chicken and set aside.
Add diced onion and minced garlic to the same pot; sauté for 3-4 minutes until the onion becomes translucent and fragrant. Scrape up any browned bits from the bottom of the pot as this adds flavor.
Stir in the chopped parsnips and celeriac, cooking for another 5 minutes until they start to soften and turn lightly golden around the edges. This step helps lock in their earthy sweetness and develops depth of flavor.
Return the seared chicken to the pot, placing it on top of the vegetables. Pour in chicken broth, then sprinkle with thyme, salt, and black pepper. Bring the mixture to a gentle simmer.
Cover the pot, reduce the heat to low, and let everything simmer for about 35-40 minutes. During this time, the chicken becomes tender and the flavors meld, while the vegetables soften to a creamy texture.
Remove the chicken from the pot; check the vegetables—they should be fork-tender and slightly caramelized around the edges. Taste the broth and adjust seasoning if needed.
Serve the stew hot, with the chicken thighs either shredded or served whole on top. Ladle the rich, earthy broth over the tender vegetables and meat.