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.