Position one oven rack as high as it can go while still accommodating the chicken atop a 10- to 12-cup Bundt pan. Heat the oven to 350°F. Cover the hole at the center of the Bundt pan with aluminum foil.
Mix 1 tablespoon of the oil, the mustard, thyme, 1/2 teaspoon of the salt, and the pepper in a large bowl until smooth.
Set the chicken in the Bundt pan, legs down, so that its large opening fits over the center of the pan and the chicken stands up in the pan. Rub the mustard mixture over the exterior of the bird.
Add the remaining 2 tablespoons oil and 1/2 teaspoon salt to that same large bowl, then add the shallots and potatoes. Toss well to coat.
Position the shallots and potatoes around the chicken, using the flutes of the pan to stand them up where possible. Set the entire Bundt pan operation on a large lipped baking sheet.
Roast until the potatoes are tender, the chicken is well browned, and an instant-read meat thermometer inserted into the thickest part of a thigh (without touching bone) registers 165°F, about 2 hours.
Let sit at room temperature for 10 minutes, then take the chicken off the top of the Bundt pan. Carve and serve with the potatoes and shallots, discarding any juices in the pan.
Recipe Notes
• For a silkier finish, substitute melted unsalted butter for the olive oil.
• Single-lobed shallots are sometimes called “French shallots.”