Preheat the oven to 450 degrees F. Put the potatoes in a saucepan and cover with cold water; season with salt. Bring to a boil, then reduce the heat to medium and simmer until tender, 15 minutes. Reserve 1/4 cup cooking water, then drain the potatoes; return to the pot. Add the sour cream, butter and reserved cooking water and mash with a masher or fork. Stir in the chives and season with salt and pepper. Cover to keep warm.
Meanwhile, place the chicken skin-side up on a rimmed baking sheet; season with salt and pepper. Bake 20 minutes; remove from the oven.
Mix the mustard, lemon juice, garlic, 2 teaspoons olive oil and the thyme; brush on the chicken and push to the sides of the pan. Toss the broccolini with the remaining 1 teaspoon olive oil in the middle of the pan; season with salt and pepper. Bake until the chicken is cooked through and the broccolini is tender, 10 minutes. Serve with the potatoes.
Photograph by Antonis Achilleos
Recipe courtesy of Food Network Magazine