Heat large sauté pan over medium heat. Add olive oil and then onion. Cook until translucent then add garlic. Do not brown garlic. Add lamb and cook until done.
Remove meat from pan and drain.
Add chicken stock and potato to small saucepan over medium heat. Bring to simmer and cook 3-5 minutes until tender. Remove potato from pan and continue to simmer chicken stock until reduced to ¼ cup. Add potato and reduced stock to lamb.
Add minced mint to lamb and salt and pepper to taste. Reserve.
Prepare favorite unsweetened pie dough such as Nathalie Dupree’s Very Beginner’s Pie Crust. Double Nathalie’s recipe for an 8 hand pie yield in this recipe. After preparing dough, allow to rest in refrigerator for at least 30 minutes.
Roll dough to ¼-inch thick and cut into 6-inch circles (Using bread plate or bowl as a guide is helpful). Place ⅓ cup filling in center of each circle. Drizzle filling with olive oil then fold dough over and seal edge.
Preheat oven to 375 degrees. Refrigerate pies for 30 minutes.
Combine egg and milk and use pastry brush to brush onto pies. Slice vent in top of each pie. Bake for 30 minutes or until golden brown.