Divide dough into 4 pieces. Roll out 1 piece of dough on a lightly floured surface to an 8" round; transfer to a parchment-lined baking sheet. Using an offset spatula, spread one-quarter of goat cheese mixture over dough, leaving a 1" border. Scatter 1 Tbsp. dill evenly over, then top with a thin layer of potato slices. Drizzle with oil; season with salt. Scatter one-quarter of leeks over potatoes. Fold edges of dough over filling, tucking and overlapping slightly as needed. Repeat with remaining dough, goat cheese mixture, chopped dill, potatoes, and leeks and more oil and salt to make 3 more galettes. Brush top of dough with egg.