For the dough: Mix together the flour, cheese, salt, pepper, and nutmeg in a large bowl. Add the butter and combine with fingertips until the mixture resembles coarse meal. In another bowl, whisk together the egg yolks and water; drizzle into the flour mixture while blending quickly with one hand until the mixture comes together into a ball. Flatten to form a disk, wrap tightly in plastic, and refrigerate for 1 hour or overnight.

Preheat the oven to 350°. Cut off one-quarter of the dough, wrap in plastic wrap and refrigerate. On a lightly floured surface, roll the remaining dough to a 13-inch round, about ⅛-inch thick. Fit into an 11-inch removable-bottom tart pan, pressing into edges; trim excess. Line shell with parchment, leaving a 1-inch overhang; fill with pie weights. Place on a baking sheet and bake until edges start to brown, 18 to 20 minutes. Remove the parchment and weights, and bake until golden, 3 to 5 minutes more. Transfer to a wire rack to cool completely.

For the filling: In a large saucepan, combine the carrots, butter, sugar, 1 teaspoon salt and ½ cup water. Bring to a simmer, cover, and cook until the carrots are very tender, about 15 minutes. Transfer the mixture to a food processor fitted with a metal blade or a blender. Add the cheeses, egg yolks, 1 of the eggs and herbs; purée until smooth. Season to taste with salt and pepper. Add filling to the tart shell.

Reduce oven temperature to 325°. Roll out the remaining piece of dough into a ⅛-inch-thick rectangle, and cut into ⅛-to-¼-inch strips. Weave a lattice of twisted dough strips on top of the filling, pressing the ends to the tart edges to seal. Whisk the remaining egg with 1 tablespoon water and brush over the lattice strips.

Place tart on a baking sheet. Bake, rotating sheet halfway through, until the custard is set and slightly puffed, 25 to 30 minutes. Transfer the tart to a wire rack and let cool for at least 10 minutes before serving.