Step 1

Pulse flour, butter, and salt in a food processor until pea-size pieces of butter form. Transfer to a medium bowl. Add 1/4 cup ice water and stir just until shaggy clumps form, adding more water by teaspoonfuls if dry. Divide dough in half; flatten each half into a disk. Wrap disks in plastic wrap and chill for at least 2 hours. DO AHEAD: Dough can be made 2 days ahead. Keep chilled.

Step 2

Preheat oven to 325°. Combine broth, 1/2 chopped onion, 1 chopped garlic clove, whole peppercorns, thyme, and bay leaves in a medium pot. Add pork shoulder; season with salt and pepper. Bring to a simmer over medium heat.

Step 3

Cover pot. Transfer to oven; braise until pork shoulder is tender and shreds easily, about 2 hours. Remove from oven; let cool.

Step 4

Transfer pork shoulder to a work surface. Shred meat with your fingers and transfer to a medium bowl. Strain pan juices through a fine-mesh sieve; add 1/2 cup juices to pork; discard solids in strainer.

Step 5

Melt butter in a large skillet over medium heat. Add remaining 1 chopped onion and 3 chopped garlic cloves; cook, stirring often, until soft, 5–7 minutes. Add mushrooms; cook, stirring often, until almost all liquid is evaporated, 5–7 minutes. Add wine; stir, scraping up browned bits. Bring to a boil; cook, stirring often, until liquid is almost evaporated, about 5 minutes.

Step 6

Add ground pork, cinnamon, and cloves. Cook, stirring to break up into small pieces, until pork is cooked through, about 5 minutes. Add potato. Cook until potato is soft, about 10 minutes. Remove from heat. Stir in shredded pork with juices. Season to taste with salt and pepper; let cool slightly. Chill until cold, about 1 hour. DO AHEAD: Can be made 1 day ahead. Cover and keep chilled.

Step 7

Roll out 1 dough disk on a lightly floured surface into a 12" round. Transfer to pie dish, leaving overhang. Fill with cooled meat mixture. Roll out remaining dough disk into a 10" round. Place dough over meat filling. Fold overhang over top crust and crimp edges. Brush crust with egg yolk. Cut three 2" slits in top crust. Chill for 1 hour.

Step 8

Preheat oven to 400°. Bake tourti&egrav;re for 30 minutes. Reduce heat to 350°; bake until crust is golden brown and filling is bubbling, 40–50 minutes. Let cool for 20 minutes before serving.