In a small bowl combine yeast, 1/4 tsp sugar and warm milk. Set aside for 5 minutes.
In a stand mixer cream together the butter and 3 Tbsp of sugar until light and fluffy, about 3 minutes.
Add the yeast mixture, egg, salt and flour and mix on low until combined.
Increase speed and mix/knead for about 7 minutes, until the dough is smooth and has pulled away from the sides of the bowl.
Press the dough into an ungreased 9x13'' baking dish. Cover with a towel or plastic wrap and let rise in a warm place until doubled, about 2 hours.
For the topping:
Whisk together light corn syrup, water and vanilla until combined.
In a separate bowl cream together the butter, sugar and salt until light and fluffy, about 5 minutes.
Add egg, scraping down the sides of the bowl. Add a little of the flour, alternating with adding the corn syrup mixture, until both are combined.
Preheat oven to 350 degrees F.
Drop large spoonfuls of topping all over the risen dough. Use a spatula to gently smooth it into an even layer.
Bake for 35-40 minutes or until the top has set and is golden brown. The center should still seem soft when it comes out of the oven. Allow to cool on a wire cooling rack to room temperature.
Serve sprinkled with powdered sugar. This cake is best enjoyed the day it is made.
Make ahead: Though I think gooey butter cake is best enjoyed the day it is made, it can be made up to 3 days ahead of time.To store: Store gooey butter cake at room temperature for 3-4 days or in the refrigerator for up to 1 week.To freeze: Allow the cake to cool to room temperature and then decide on a freezing method. I like to cut gooey butter cake into smaller bars and place them in a gallon freezer ziplock bag. This way, I can thaw just one piece at a time. You can also freeze the whole cake by covering the pan (as long as it's aluminum and not glass) in plastic wrap or tinfoil and freeze for up to 3 months.