Center a rack in the oven and preheat the oven to 425°F Line a baking sheet with a silicone baking mat or parchment paper.
Cut the carrots and leeks into slender batons or sticks: First cut the carrots lengthwise in half, then place the halves cut side down on the cutting board and cut crosswise in half or cut into chunks about 3 inches long. Cut the pieces into 1/8- to ¼-inch-thick matchsticks. If your carrots were fat and you think your matchsticks don’t look svelte enough, cut them lengthwise in half. Cut the leeks in the same way.
Fit a steamer basket into a saucepan. Pour in enough water to come almost up to the steamer, cover, and bring to a boil. Drop the carrots, leeks, and 1 rosemary sprig into the basket, cover, and steam until the vegetables are tender enough to be pierced easily with the tip of a knife, 10 to 15 minutes. Drain the vegetables and pat them dry; discard the rosemary sprig.
In a medium bowl, whisk the eggs together with the crème fraîche or heavy cream. Add the mustard, season with salt and white pepper — mustard has a tendency to be salty, so proceed accordingly, and whisk to blend. Taste and see if you want to add a little more of one or the other mustards.
Put the tart pan on the lined baking sheet and pour the filling into the crust. Arrange the vegetables over the filling — they can go in any which way, but they’re attractive arranged in spokes coming out from the center of the tart. Top with the remaining rosemary sprig and give the vegetables a sprinkling of salt and a couple of turns of the peppermill.
Bake the tart for about 30 minutes, or until it is uniformly puffed and lightly browned here and there and a knife inserted into the center of the custard comes out clean. Transfer the tart to a cooling rack and let it rest for 5 minutes before removing the sides of the pan. Serve hot, warm, or at room temperature (or lightly chilled).