Whisk flour, sugar, and 1 teaspoon salt in a bowl. Using a dough blender, two forks, or your fingers, cut butter into flour mixture, forming pea-size crumbles. Add 1 egg and water and work dough until smooth. (Alternatively, pulse ingredients in a food processor.) Flatten dough into a disk and wrap in plastic wrap. Refrigerate 1 hour before using.
Heat oven to 400°. On a lightly floured surface, roll dough into a 12-inch round. Fit into a 9-inch tart pan with a removable bottom. Trim edges chill 30 minutes. Using a fork, prick the dough all over. Line dough with parchment paper and fill with pie weights or dried beans. Bake for 20 minutes. Remove paper and weights and bake an additional 5 minutes, then cool.
Meanwhile, julienne the carrot and the leek into matchsticks. Fill a large wok with 2-inches water and place over medium-high heat. Fit a steamer basket on top and add the carrots, leeks, and rosemary sprigs. Cover the steamer and cook until the vegetables are tender, about 10 minutes. Remove the rosemary and allow the vegetables to cool completely.
In a medium bowl, whisk the remaining three eggs with the creme fraiche, mustards, salt, and pepper. Pour filling into tart and arrange the vegetables over the top. Bake until the tart is lightly browned and set, about 30 minutes. Let cool for 5 minutes before serving, or serve at room temperature.