For the cake:
3 tbsp. milk, at room temperature
2 tbsp. warm water (105-110˚ F)
1¾ tsp. active dry or instant yeast
6 tbsp. unsalted butter, at room temperature
3 tbsp. sugar
1 tsp. salt
1 large egg
1¾ cups all-purpose flour
For the gooey butter topping:
3 tbsp. plus 1 tsp. light corn syrup
2 tbsp. water
2½ tsp. vanilla extract
12 tbsp. (¾ cup) unsalted butter, at room temperature
1½ cups sugar
1 tsp. kosher salt
1 large egg
1 cup plus 3 tbsp. all-purpose flour
Confectioners’ sugar, to serve
Butter a 9 x 13-inch baking dish. To make the cake dough, combine the milk, water, and yeast in a small bowl or liquid measuring cup. Mix to dissolve the yeast in the water and set aside to let foam slightly, a few minutes. In the bowl of an electric mixer, combine the butter, sugar and salt. Beat on medium-high speed until light, about 2 minutes. Blend in the egg. Beginning and ending with the flour, alternately add the flour and the milk mixture, until a smooth dough is formed. Transfer the dough to the prepared pan and stretch and nudge the dough until it covers the bottom of the pan in an even layer. Cover the pan with a clean kitchen towel or plastic wrap and let rise in a warm place until puffed and about doubled in volume, approximately 2-3 hours.
Preheat the oven to 350˚ F. To make the topping, combine the corn syrup, water and vanilla in a small bowl and whisk to blend. With an electric mixer, beat together the butter, sugar and salt on medium-high speed until light and fluffy, about 5 minutes. Scrape down the sides of the bowl. Blend in the egg. Alternately add the flour and the corn syrup mixture until just smooth, scraping down the bowl as needed.
Drop large dollops of the topping over the surface of the dough and gently spread the topping into an even layer. Bake until light golden brown, about 25-30 minutes. The edges will be brown and set, but the center will still be light and gooey. Allow to cool briefly in the pan before slicing and serving, at least 30 minutes. Dust with confectioners’ sugar before serving.