Butter a 9-inch springform pan.
Put milk and butter in a heatproof dish and microwave until butter is melted and milk is warm. If hot, let cool until warm to the touch.
Combine ½ cup flour, sugar, yeast and salt in a medium bowl. Whisk to combine. Pour milk mixture into dry ingredients and beat with a handheld mixer to combine. Add egg and beat until smooth. Mix in remaining ½ cup of flour. Mixture will resemble a stiff batter. Scrape into prepared pan and smooth to cover the surface of the pan. Cover loosely with a dish towel and let rise in a warm spot until doubled in size, about 45 minutes. Towards the end of rise time, preheat oven to 350°F and make the filling.
Use a handheld mixer to cream butter, sugar and salt in a medium bowl until light and fluffy. Add egg and mix, scraping down side of bowl as needed. Beat in corn syrup. Add water and vanilla and mix. Add flour and slowly mix until thoroughly combined. Spoon topping over risen cake batter, then gently spread it in an even layer.
Bake for 25-30 minutes, or until the edges are set and the center is still jiggly when shaken. Let cool 5 minutes.
Run a knife around the sides of the pan, then release and remove the rim of the springform pan. Serve gooey butter cake with a dusting of powdered sugar.