Bring a large pot of water to a boil. Drop the potatoes in and boil 8 minutes, until tender. Drain the potatoes and spread them out onto a baking sheet. Chill until cold.
Preheat oven to 350 degrees. Whisk together both mustards, honey, brown sugar and thyme. Place the chicken in a 9x13 baking dish and baste with the honey mustard mixture. Roast for 30 minutes.
Baste the chicken again and roast for another 15-20 minutes.
Make the vinaigrette: whisk the vinegar, 2 Tbsp. oil, 1 tsp. Dijon, sugar and shallots in a small bowl. Set aside.
Once the potatoes are cold, heat remaining 2 Tbsp. oil in a cast iron skillet. Saute the potatoes until brown and crisp. Toss the cooked potatoes with the vinaigrette and arugula.