1 2/3cupsunbleached all-purpose flour, plus additional for work surface3/4 teaspoon table salt10 tablespoons unsalted butter, cold, cut into 1/2-inch pieces and frozen for 10 minutes2 tablespoons sour cream4 – 6 tablespoons ice water
For the Chicken and Vegetables:
1teaspoonvegetable oil1 whole chicken, about 4 1/2 pounds, cut into 4 pieces (2 breast pieces, 2 leg quarters, wings discarded) and trimmed of excess fat24 frozen pearl onions, thawed, drained, and patted dry, about 1 cup1 medium onion, cut into 1/2-inch pieces (about 1 cup)4 medium carrots, cut into 1/2-inch pieces (about 1 1/2 cups)3 medium ribs celery, cut into 1/2-inch pieces (about 1 cup)Salt and ground black pepper
For the Sauce:
4 1/4cupslow-sodium chicken broth4 tablespoons unsalted butter1/2 cup unbleached all-purpose flour3/4 cup heavy cream3/4 cup frozen peas, unthawed2 teaspoons fresh lemon juice1 teaspoon minced fresh parsley leaves1 teaspoon minced fresh thyme leaves
To Make the Dough: In the bowl of a food processor, process flour and salt together until combined, about 3 seconds. Add butter and pulse until size of large peas, about six to eight 1-second pulses.
In small bowl, mix sour cream and 4 tablespoons ice water until combined. Add half of sour cream mixture to flour mixture and pulse for three 1-second pulses. Repeat with remaining sour cream mixture. Pinch dough with fingers; if dough is floury, dry, and does not hold together, add 1 to 2 additional tablespoons water and process until dough forms large clumps and no dry flour remains, three to five 1-second pulses.
Turn dough onto work surface and shape into ball. Flatten into a 5-inch disk and wrap in plastic. Refrigerate until firm but not hard, 1 to 2 hours. (Dough can be refrigerated overnight.)
To Roast the Chicken and Vegetables: Heat oven to 450 degrees. In heavy-bottomed 12-inch ovenproof skillet, heat oil over medium-high heat it just begins to smoke; swirling to coat evenly with oil. Brown chicken pieces skin side down until deep golden, 3 to 4 minutes; turn chicken pieces and brown until golden on second side, 3 to 4 minutes longer. Transfer chicken to large plate; drain all but 1 tablespoon fat.
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.