Preheat oven to 325°F (170°C).

In the bowl of a stand mixer fitted with the paddle attachment, beat butter and granulated sugar at medium speed until light and fluffy, about 5 minutes, stopping to scrape bottom and sides of bowl. Add eggs and egg white, one at a time, beating well after each addition.

In a large bowl, whisk together 5 cups (625 grams) flour, salt, baking powder, and cardamom. In a small bowl, stir together buttermilk and vanilla bean paste. With mixer on low speed, gradually add flour mixture to butter mixture alternately with buttermilk mixture, beginning and ending with flour mixture, beating just until combined after each addition.

In another medium bowl, toss together Honey-Poached Plumcots and remaining 2 tablespoons (16 grams) flour to coat.

Spray a 15-cup Bundt pan with baking spray with flour. Spoon one-fourth of batter (about 2½ cups or 558 grams) into prepared pan, spreading smooth with an offset spatula. Firmly tap pan on a kitchen towel-lined counter several times to release any air bubbles. Sprinkle one-third of plumcots mixture (about ⅔ cup or 147 grams) onto batter in pan, keeping fruit about ¼ inch away from sides of pan. Repeat layers of batter and plumcots two more times; top with remaining batter, spreading smooth.

Bake until a wooden pick inserted near center comes out clean and an instant-read thermometer inserted near center registers at least 205°F (96°C), 1 hour and 45 minutes to 2 hours. Let cool in pan on a wire rack for 10 minutes. Invert cake onto a wire rack, and let cool completely.

In another small bowl, whisk together confectioners’ sugar and poaching liquid until smooth; spoon onto cooled cake. Refrigerate in an airtight container for up to 3 days.

In a medium saucepan, bring all ingredients to a boil over medium-high heat, stirring occasionally. Reduce heat, and simmer until fruit is just tender, 8 to 10 minutes. Remove plumcots using a slotted spoon, and let cool completely on a wire rack. Let poaching liquid cool completely.