Preheat oven to 375°F. Melt butter in a large skillet over medium-high. Add leeks, carrots, thyme, 1 ½ teaspoons salt, and ½ teaspoon pepper. Cook, stirring occasionally, until vegetables have softened, about 5 minutes.
Sprinkle with flour; cook, stirring, 1 minute. Whisk in broth and add potato. Bring to a boil, then reduce heat and simmer, partially covered and stirring occasionally, until potato is tender and mixture has thickened, 6 to 7 minutes. Add turkey. Let cool about 30 minutes.
On a lightly floured surface, roll dough into a 15-inch round; fit into a deep-dish pie dish. Add stuffing and dollop with half of cranberry sauce, then add turkey mixture. Fold dough over, leaving center exposed. Brush dough with egg.
Bake 40 minutes. Dollop top with remaining cranberry sauce; bake until bubbling in center and bottom of crust is golden, 30 to 40 minutes more. Let cool at least 30 minutes before serving.
