Position a rack in the upper third of the oven and preheat the oven to 425°.
Lightly oil a roasting pan.
Toss the potatoes, carrots, turnip, and parsnip in a large bowl with the oil, 1 teaspoon of the thyme, 3/4 t. salt, adn 1/2 t. pepper.
Spread in the roasting pan.
Season the chicken with the remaining 3/4 t. salt and 1/2 t. pepper, then sprinkle with the remaining 1 t. thyme.
Nestle the chicken, skin side up, among the vegetables.
Roast, occasionally scraping up and turning the vegetables with a metal spatula, until the vegetables are tender and an instant-read thermometer inserted in the thickest part of the chicken breasts reads 170°, about 35 minutes.
Transfer the chicken and vegetables to a platter.
If the chicken is tender before the vegetables are done, tent the chicken with foil, then continue roasting the vegetables.
In either case, tent the platter while making the pan sauce.
Place the roasting pan over 2 burners on high heat.
Add the broth and bring to a boil, scraping up the browned bits in the bottom and sides of the pan.
Boil until reduced to about 1/3 cup, about 5 minutes.
Remove from the heat, and whisk in the butter to lightly thicken the sauce.
Pour over the chicken and vegetables and serve hot.