1. Brush a 13-by-9-inch glass baking dish with butter. Line with parchment paper, leaving a 2-inch overhang on each of the long sides, and butter. Set aside.

2. Make the cake: In a small bowl, stir together milk and yeast; let stand until slightly foamy, about 5 minutes.

3. In the bowl of an electric mixer fitted with the paddle attachment, beat butter, sugar, and salt until light and fluffy, about 2 minutes. Add eggs, one at a time, beating well after each addition. Add flour in two additions, alternating with the milk mixture. Beat on medium speed until it forms a soft dough, 3 to 5 minutes.

4. Press dough into prepared baking dish and cover loosely with plastic wrap. Set aside in a warm place until puffy and full of air, about 1 hour.

5. Make the topping: Preheat oven to 350 degrees. In the bowl of an electric mixer fitted with the paddle attachment, cream the butter, sugar, and salt on medium-high speed until light and fluffy, about 2 minutes. Add egg, egg yolk, and vanilla and beat to combine, scraping down sides of bowl. Add flour in two additions, alternating with corn syrup. Beat until thoroughly combined.

6. Spread topping evenly over risen dough using an offset spatula. Bake until just golden around the edges, about 20 minutes. Let cool completely in baking dish on a wire rack before dusting with confectioners' sugar and serving.