Preheat the oven to 375 degrees F.
Season the chicken thighs with salt and pepper, to taste. In a medium oven-proof frying pan over high heat, add the oil and butter and stir to combine. Add garlic and thyme and cook until the butter is slightly browned. Add the chicken, skin side, down and let it sit in the pan so the skin becomes crispy and golden. While the skin is crisping up, spoon the hot butter/oil over the thighs. Once the skin is crisp, about 5 minutes, and releases from the pan easily, turn the chicken over. Continue cooking for another 5 minutes to sear the meat. Put in the oven and roast for approximately 10 to 15 minutes. Depending on how large the thighs are, check for doneness.
Have an ice bath ready before beginning. Bring a medium stockpot of salted water to boil over medium heat. Blanch each of the varieties of potatoes and carrots; starting with the fingerlings, Yukon gold, carrots and purple Peruvians last. Cook each batch until tender, but still with a little bite. After blanching each set of vegetables, use a strainer remove all of the vegetables out of the water and dunk them in the ice water bath. Once they are cold to the touch, drain and set aside. Right before serving, get a medium fry pan heated over high heat, add the a little oil and some butter. Once the butter is melted and hot, add the potatoes and carrots and saute until heated through. Season with salt and pepper, to taste.
For the Herbed Jus:
In a medium saucepan over medium heat, add the oil and 1 tablespoon butter. When the butter has melted, add the onions, garlic and shallots. Cook slowly to soften and bring out the flavors. Do not burn the garlic. Reduce the heat to low if the mixture starts to brown. Stir in the Sauvignon Blanc and allow it to reduce by half. Add the chicken stock and continue cooking at a simmer to reduce again by half. Taste the jus and if it seems a bit too tart and sharp, add more butter to smooth it out. (If you want to make the jus into a thicker sauce, you can add some heavy cream.) Adjust the seasoning with salt and pepper, to taste. When the jus is ready, add the 1/4 cup parsley and 1/2 cup chives right before serving to maintain the bright green color.
Put a large serving spoon of heated, seasoned potatoes and carrots in the center of each serving plate, letting some of them spill to the edges. Arrange 1 chicken thigh on top of the vegetables, skin side up. Ladle the jus around each plate and garnish the top of the chicken with a small sprinkling of the minced parsley and chives.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.