Preheat oven to 350°F (180°C). Butter a 13x9-inch baking pan. Line pan with parchment paper, letting excess extend over sides of pan.

In a medium bowl, whisk together milk, yogurt, eggs, egg yolks, and vanilla bean paste.

In the bowl of a stand mixer fitted with the paddle attachment, beat flour, sugar, baking powder, and salt at low speed just until combined. Add butter, one cube at a time, beating until combined after each addition. (Mixture should resemble sand.) Add half of milk mixture; beat at low speed just until combined. Add remaining milk mixture, and beat at medium speed just until combined. Using a rubber spatula, fold batter to ensure that no dry ingredients remain. Pour batter into prepared pan, smoothing top with an offset spatula.

Bake until cake springs back when touched in center and a wooden pick inserted in center comes out clean, about 45 minutes. Let cool completely in pan.

Using excess parchment as handles, remove cooled cake from pan. Using an offset spatula, spread Vanilla Bean Swiss Meringue Buttercream onto cake, swirling as desired. Garnish with sprinkles, if desired.