In a food processor, pulse the flour and 1 tsp. salt to combine. Cut 2 sticks butter into small cubes. Add to the flour and pulse until pea-size pieces form. Drizzle in the ice water; process just until the dough starts to form a ball on the blade. Knead briefly to blend. Halve the dough; pat into 2 disks.
Preheat the oven to 425 degrees. On a floured surface, roll out each dough half into a large rectangle or circle. Transfer to two 5-by-14-inch rectangular or 9-inch round tart pans; press in evenly. Prick the dough all over with a fork; refrigerate for 10 minutes. Cover the dough with parchment and fill with pie weights. Bake for 10 minutes. Remove the parchment and pie weights; reduce the oven to 375 degrees. Bake until golden, 15 to 18 minutes. Transfer the crust to a rack to cool completely.
In a large skillet, melt 2 tbsp. butter over medium-high heat. Add half of the mushrooms and cook until most of their liquid has evaporated and the mushrooms are soft and brown, about 10 minutes; season and transfer to a bowl. Repeat with 2 tbsp. more butter and the remaining mushrooms. Melt the remaining 2 tbsp. butter in the skillet. Add the leeks and cook until softened and just barely browned, 6 to 8 minutes; season.
In a small bowl, whisk the goat cheese, milk, eggs and a generous pinch of pepper. Divide the goat cheese mixture between the crusts; spread to cover. Divide the leeks between the tarts, sprinkling evenly on top of the cheese. Cover with the mushrooms. Return the tarts to the oven; bake just until the cheese mixture sets, about 15 minutes.
Sprinkle the tarts with parsley and thyme. Serve within 3 hours.