Is this delicious yeasted cake dessert, or breakfast? With a yeasted base topped with vanilla-y, buttery goo, that's up to you.
Dissolve the yeast in the milk and water. (You can add yeast with the flour if desired, but there is a chance that the yeast will not mix into the dough very well.)
With a stand or electric mixer, beat together the butter, sugar, and salt for a few minutes, until fluffy. Scrape the bowl, then add egg and mix until incorporated.
Add half of the flour and yeast, then all of the milk and water, then the remaining flour and yeast, mixing well after each addition. Scrape the bowl, then mix at medium speed for 4-5 minutes. At this point, the dough will be smooth and stretchy. (Note: if using a hand mixer, don't overwork it. If you can't mix the dough, just pull it out and knead it for a few minutes by hand. It's a very easy to handle dough.)
About 20 minutes from the end of rising time, preheat the oven to 350F and make the gooey topping.
Using a stand or electric mixer, cream together the butter, sugar, and salt until light and fluffy. Scrape the bowl, then add the egg and mix until combined.
In a small bowl or measuring cup, whisk together the corn syrup, water, and vanilla.
To the butter mixture, add half the flour, then the corn syrup mixture, then the rest of the flour. Scrape down the bowl, then mix for 30 seconds more. Spoon and spread the topping over the cake dough.
Bake cake for 30 (in a metal pan) to 45 (in a ceramic or glass pan) minutes. The cake will still be liquidy in the center when it is done, but it will be golden around the edges. You will think it's not done, but it is. (This is less done than in the pictures of the full pan I have above.)
Remove the cake from the oven and cool completely in the pan. The filling will initially be puffy but will deflate as it cools. Dust with powdered sugar before serving. Store in the refrigerator.
This recipe is adapted from King Arthur Flour.