Heat oven to 425°F. Toss whole potatoes with 1 tablespoon oil and 1/4 teaspoon salt on 18-by-13-inch rimmed sheet. Roast 35 to 40 minutes until centers easily pierced.
Mix mustard, vinegar, garlic, pepper, rosemary, and 1/4 teaspoon salt. Coat thighs; refrigerate while potatoes finish first roast.
Press hot potatoes with sturdy mug bottom to about 1/2 inch thick. Drizzle with remaining oil; add remaining salt.
Space thighs between potatoes; roast 25 to 30 minutes until chicken reaches 165°F and potato edges crisp.
Rest 5 minutes; serve each thigh with one-quarter potatoes.