How to Make It
Melt butter in a large Dutch oven over medium. Add onion, thyme, sage, and rosemary; cook, stirring occasionally, until onion is softened and translucent, 8 to 10 minutes. Add lamb pieces; cook, stirring occasionally, until browned on all sides, 10 to 12 minutes. Add flour; cook, stirring constantly, until flour is lightly browned and mixture is thickened, 1 to 2 minutes. Gradually stir in broth. Bring to a boil over medium-high. Reduce heat to medium-low; simmer, uncovered, stirring occasionally, 1 hour and 30 minutes. Continue cooking mixture, stirring often, until lamb is fall-apart tender and liquid is reduced to a very thick gravy, about 1 hour, adding water, 1/4 cup at a time, if sauce becomes too thick before lamb is tender. (Lamb will begin to shred when stirred toward end of cook time.) Coarsely shred lamb (still in Dutch oven) using 2 forks. (You will have about 3 cups filling once lamb is shredded.)
Season filling with salt to taste. Transfer mixture to a large shallow heatproof bowl; let cool completely, about 1 hour. (Cooled filling can be used immediately or stored in an airtight container in refrigerator up to 3 days. If chilled, let filling stand at room temperature 30 minutes before assembling pies.)
Stir together flour and salt in a large bowl. Add butter pieces; rub together using your fingers until evenly incorporated and mixture resembles small peas. Beat 3 eggs in a medium bowl. Make a well in the center of flour mixture; add beaten eggs, and stir with a fork to combine. Stir in 3 tablespoons ice water. Continue adding up to 2 tablespoons ice water, 1 tablespoon at a time, until a shaggy dough forms. Wrap dough tightly in plastic wrap; refrigerate at least 3 hours or up to 3 days.
Unwrap chilled dough; place on a lightly floured work surface, and roll to 1/4-inch thickness. Using a 5-inch round cutter, cut 10 dough rounds; place rounds on a baking sheet lined with parchment paper. Using a 4-inch round cutter, cut 10 dough rounds, rerolling scraps as needed. Add to baking sheet. Cover baking sheet tightly with plastic wrap; chill at least 20 minutes or up to 1 day.
Preheat oven to 350°F. Arrange the 4-inch dough rounds at least 1 1/2 inches apart on 2 baking sheets lined with parchment paper. Beat remaining egg in a small bowl. Spoon about 1/3 cup filling onto center of each 4-inch round; brush dough edges lightly with beaten egg. Top with 5-inch dough rounds. Using your hands, cup top dough round over filling, pressing lightly to shape dough around filling. Using a fork, crimp edges to seal. Cut a 1/4-inch-round hole in top of each pie.
Brush pies lightly with remaining beaten egg. Bake in preheated oven until golden brown, 32 to 38 minutes. Transfer pies to a wire rack; let cool 10 minutes. Serve hot.