Mix flour and salt together in a bowl. Cut in 1/2 cup butter with a pastry blender until mixture resembles small pebbles.
Mix 1/4 cup ice water, yogurt, and lime juice together in a small bowl. Stir into the flour mixture until dough comes together. Shape dough into a disc; wrap in plastic wrap. Chill until firm, about 45 minutes.
Heat olive oil and 1 tablespoon butter in a large skillet over medium heat. Add leeks; cook and stir until softened, about 5 minutes. Stir in marjoram; cook until fragrant, about 1 minute. Stir in shiitake mushrooms and button mushrooms; cook and stir until tender, about 5 minutes. Remove from heat; season with salt and pepper.
Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Roll out dough on a floured work surface with a floured rolling pin. Cut into 5-inch circles. Place circles on the baking sheet.
Place 1 tablespoon of the leek and mushroom filling in the center of each circle, leaving a 1-inch border. Fold up and overlap edges of border around the filling.
Whisk egg yolk and 1 tablespoon water together in a small bowl to make egg wash. Brush egg wash over dough.
Bake in the preheated oven until light golden brown, 15 to 25 minutes.