2 1/4 cups unbleached all-purpose flour
1/4 teaspoon ground allspice
1/8 teaspoon ground cardamom
Fine sea salt
11 tablespoons (1 stick plus 3 tablespoons) unsalted butter, softened
1/4 cup plus 1 1/2 teaspoons granulated sugar
1 vanilla bean, split lengthwise, seeds scraped out and reserved
3 large egg yolks
CONFIT AND SYRUP
1 cup water
1 cup granulated sugar
5 ounces rhubarb stalks, cut crosswise
into 1 1/2-inch pieces (about 1 1/4 cups)
1/2 pound carrots, peeled and cut into 1/4-inch rounds (about 1 1/2 cups)
1 1/4 cups sliced almonds
1/2 cup plus 1 tablespoon unbleached all-purpose flour
Fine sea salt
14 ounces rhubarb stalks, cut in half lengthwise then cut crosswise into
1-inch pieces (about 3 1/2 cups)
1/3 cup plus 1 tablespoon granulated sugar
8 tablespoons (1 stick) unsalted butter, softened
1 cup confectioners sugar
2 large eggs
2 large egg yolks
1/4 cup heavy cream
3 tablespoons whole milk
SPECIAL EQUIPMENT: a pasta pot with a deep perforated colander-steamer insert or a metal steamer; a 9-inch springform pan
FOR CRUST: In a large bowl, whisk together flour, allspice, cardamom and generous pinch salt. In the bowl of an electric mixer fitted with the paddle attachment, combine butter, sugar and vanilla seeds; mix on medium speed until well combined, about 4 minutes. Add egg yolks, one at a time, allowing each to incorporate before adding the next. In 2 additions, add flour mixture, scraping down sides of bowl between additions.
Turn out dough onto clean work surface. Shape into flattened disc and wrap in plastic wrap. Chill in refrigerator at least 1 hour or overnight.
FOR CONFIT AND SYRUP: Set a wire rack over a rimmed baking sheet.
In a medium saucepan, combine water and sugar; bring mixture to a boil over high heat. Add rhubarb; return mixture to a boil, then remove pan from heat. Let rhubarb stand in syrup 5 minutes. Using a slotted spoon, transfer rhubarb pieces to a plate. Return pan to medium-high heat; boil syrup until reduced by half, 5 to 7 minutes.
Add rhubarb pieces back to syrup; return mixture to a boil. Remove pan from heat and let rhubarb stand in syrup for 5 minutes. Using slotted spoon, transfer rhubarb to plate. Return pan to medium-high heat; boil syrup until reduced by 1/4, about 4 minutes.
Add rhubarb pieces back to syrup and return mixture to a boil. Remove pan from heat. Using slotted spoon, transfer rhubarb confit pieces to prepared wire rack; set aside. Reserve syrup.
FOR CAKE: Bring a few inches of water to a boil in pot so that bottom of steamer insert sits above water. Arrange carrots in insert. Steam, covered, over gently simmering water until very tender, about 15 minutes. Transfer carrots to the bowl of a food processor or blender; purée until smooth. Transfer carrot purée to a bowl.
In the bowl of a food processor, combine almonds and 1 tablespoon flour; grind until nuts are very finely chopped (to make almond flour). Transfer almond flour to a large bowl. Add remaining all-purpose flour and generous pinch salt; whisk to combine.
Heat oven to 350º with rack in middle. On a well-floured surface, roll out dough to a 16-inch round. Fit dough into 9-inch springform pan, pressing dough into edges.
In a large skillet, combine rhubarb and granulated sugar; cook over medium heat, stirring occasionally, until rhubarb is tender, about 12 minutes. Using a slotted spoon, transfer rhubarb to a plate to cool; discard liquid in skillet.
In the bowl of an electric mixer fitted with the whisk, beat together butter and confectioners sugar on medium speed for
7 minutes. Add eggs, one at a time, then egg yolks, one at a time, allowing each addition to incorporate before adding the next. Add flour mixture, cream and milk; beat batter just to combine.
In a bowl, stir together 1 1/4 cups batter and rhubarb. To bowl with carrot purée, add 3/4 cup of the remaining batter and stir to combine. Put remaining plain batter into cake pan; spread evenly with a butter knife or offset spatula. Put carrot batter into cake pan; spread evenly to create a second layer. Put rhubarb batter into cake pan; spread evenly to create a third layer.
Put cake pan on a baking sheet. Bake, rotating pan once halfway through, until cake is set, about 40 minutes.
Reduce oven temperature to 325º, tent cake pan with foil and continue to bake until cake springs back to gentle touch, about 20 minutes more.
Remove foil and transfer cake pan to wire rack to cool for 5 minutes. Meanwhile, heat syrup to a simmer, then remove from heat. Release cake from pan and brush with 1/4 cup syrup. Let cake cool completely, then decorate top with 4 pieces rhubarb confit. Serve cake with remaining confit.