Gooey Butter Cake

Cake - Grease a 9-by-13-inch baking pan with butter; set aside.

Cake - In a small saucepan, combine milk, water, and 1/4 teaspoon sugar. Set over medium-low heat until mixture is warm to the touch, about 100°F (38°C). Remove from heat and stir in yeast to dissolve. Let sit until mixture is foamy, about 5 minutes.

Cake - In the bowl of a stand mixer fitted with the paddle attachment, combine butter, remaining sugar, salt, vanilla, and nutmeg. Mix on low speed to roughly incorporate, then increase speed to medium-high and beat until light and fluffy, pausing halfway through to scrape down the sides with a flexible spatula, 2 to 3 minutes.

Cake - Using a flexible spatula, scrape down the sides and bottom of the bowl. Add egg and egg yolk and beat on low speed to incorporate. (The mixture will slosh around a bit and look like runny scrambled eggs after 1 minute; this is normal.) Increase speed to medium and beat until batter breaks down into smaller curds, another 2 to 3 minutes. (The mixture will won't be completely emulsified and will resemble cottage cheese; this is normal.)

Cake - Scrape down the sides again and add the flour. Pulse mixer on and off until most of the flour has been absorbed, then reduce to low speed and mix until dough resembles a stiff cookie dough and mixer begins to strain slightly. With the mixer still on low speed, drizzle yeast and milk mixture into the bowl, and mix until milk is fully incorporated and batter is thick.

Cake - Carefully scrape batter off paddle into the bowl and switch to the hook attachment. Knead dough on medium-low until dough is smooth and elastic and clings to the dough hook when lifted up, 7 to 10 minutes. Using a dough scraper, scrape dough into prepared baking pan. Using ungreased hands, press the dough evenly into a 1/2-inch thick layer all the way across the bottom of the pan. Loosely cover with plastic wrap and let sit at warm room temperature (75 to 80°F; 24 to 27°C) until puffy and doubled in size, 1 1/2 to 2 1/2 hours.

Topping - About 15 minutes before cake is done rising, adjust oven rack to middle position and preheat to 350°F (175°C). In a heat-proof spouted measuring cup, combine corn syrup with hot water. In the clean bowl of a stand mixer fitted with a paddle attachment, beat butter, granulated sugar, brown sugar, vanilla, and salt until mixture is fluffy, about 2 minutes.

Topping - Using a flexible spatula, scrape down bottom and sides of the bowl and, with the mixer running on low speed, add the egg, followed by the corn syrup mixture, and mix until fully incorporated.

Topping - Scrape down bottom and sides of the bowl again and add the flour. Pulse mixer until the flour is mostly absorbed, 3 to 4 pulses, then mix on low speed until batter is thick, about 2 minutes.

Bake - Scrape batter over the risen dough (don't worry if the dough deflates slightly from the weight of the topping) and, using an offset spatula, spread the topping from side to side to evenly cover dough. Bake until top of cake is golden brown but still liquid and jiggly, and an instant-read thermometer inserted in the center registers 190° to 195°F (88° to 90°C), 35 to 45 minutes.

Bake - Remove from the oven. The top will be rippled with irregular peaks and valleys. Let the cake cool completely on a wire rack, about 1 hour. Dust with confectioners' sugar, slice, and serve.