Heat oven to 400°F. Pat the chicken dry inside and out with paper towels. Remove giblet package, if present. In a small bowl stir together mustard and oil. Spread all over chicken. Place chicken in a roasting pan or heatproof baking dish.
Roast, uncovered, for 35 minutes. Add Brussels sprouts to pan or dish; sprinkle lightly with salt and toss to coat. If chicken is browning too fast, cover loosely with foil. Roast another 25 to 35 minutes or until an instant-read thermometer registers 170°F. in the breast or 180°F. in the thigh and sprouts are tender, stirring sprouts once more during cooking.
Remove chicken from oven and let stand 10 minutes.