Set skillet over medium heat. Add pearl onions and cook, stirring occasionally, until browned, about 3 minutes. Increase heat to medium-high and add onion, carrots, celery, and salt and pepper to taste; cook, stirring occasionally, until vegetables begin to brown, 2 to 3 minutes. Return chicken pieces to skillet skin side up and place in oven. Roast until thickest part of breast registers about 160 degrees on instant-read thermometer and thickest part of thigh and drumstick registers about 175 degrees, 18 to 25 minutes. Using an oven mitt, remove skillet from oven. Transfer chicken to platter and let rest 1 hour. Set skillet with vegetables aside.

To Roll Out the Dough: Allow dough to stand at room temperature for 15 minutes before rolling if dough has been chilled longer than 2 hours. Dust 16 by 12-inch sheet of parchment paper liberally with flour. Roll out dough to rough 14-inch round on parchment. Mark center of top, bottom, and side arcs with knife. Connecting the marks, trim dough into a 14-inch circle.

Fold dough to form 1/2-inch perimeter rim.

Crimp folded dough between knuckle and forefingers to form 12-inch round with fluted edge. Cut four oval-shaped vents, 3 inches long and about 1/2 inch wide. Decorate dough with cutouts.