Preheat oven to 350°F (180°C). Spray a tall-sided 8-inch round cake pan with baking spray with flour. Line bottom of pan with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter and granulated sugar at medium speed until pale and fluffy, 3 to 4 minutes, stopping to scrape sides of bowl and paddle. Add eggs, one at a time, beating until combined after each addition. Beat in 1 teaspoon (4 grams) vanilla.

In a medium bowl, whisk together flour, baking powder, salt, and baking soda. In a small bowl, whisk together sour cream and milk. With mixer on low speed, gradually add flour mixture to butter mixture alternately with sour cream mixture, beginning and ending with flour mixture, beating until just combined after each addition and stopping to scrape sides of bowl and paddle. Using a rubber spatula, gently stir in mango. Spread batter into prepared pan.

Bake until a wooden pick inserted in center comes out with a few moist crumbs, 50 to 55 minutes, covering with foil during final 10 minutes of baking to prevent excess browning. Let cool in pan on a wire rack for 10 minutes. Remove from pan, and let cool completely on a wire rack.

In a small bowl, whisk together confectioners’ sugar, lime juice, and remaining 1 teaspoon (4 grams) vanilla. Pour onto cooled cake. Garnish with mango, if desired. Let stand until glaze is set, about 15 minutes. Refrigerate in an airtight container for up to 3 days.