Heat the oven to 425°F. Toss whole potatoes with 1 tablespoon oil and 1/4 teaspoon salt on an 18-by-13-inch rimmed sheet pan and roast for 35 to 40 minutes until a knife slides easily through their centers.
Mix mustard, balsamic vinegar, garlic, pepper, rosemary and 1/4 teaspoon salt in a bowl. Coat the chicken thighs and keep them refrigerated while the potatoes finish their first roast.
Press each hot potato with the bottom of a sturdy mug until about 1/2 inch thick. Drizzle with the remaining oil and sprinkle with remaining salt.
Space chicken thighs between the flattened potatoes and roast for 25 to 30 minutes until chicken reaches 165°F and potato edges are crisp.
Rest for 5 minutes and serve each thigh with one-quarter of the smashed potatoes.