I grew up outside of St. Louis and a local bakery made this yummy St. Louis traditional cake. I had looked for years for a recipe to make it (since I no longer live near there)...and finally found it in the New York Times (of all places) a year ago. It's an amazing version -- very authentic! It's time consuming, but worth it.
Start by making the cake dough. In a small bowl, mix milk with 2 tablespoons warm water. Add yeast and whisk gently until it dissolves. Mixture should foam slightly.
Using an electric mixer with paddle attachment, cream 6 T butter, 3 T sugar and 1 t salt. Scrape down the sides of the bowl and beat in one egg. Alternately add flour (1 3/4 cup total) and milk/yeast mixture, scraping down sides of bowl between each addition. Switch to a dough hook on the mixer and beat dough on medium speed until it forms a smooth mass and pulls away from the sides of the bowl, 7-10 minutes.
Press, stretch and nudge dough into an ungreased 9" x 13" baking dish (I actually use 2 8" square aluminum disposable baking pans, so I keep one cake for us and give the other cake away). Cover the dish with plastic wrap, put in a warm place and allow it to rise until doubled, 2 1/2-3 hours.
When you are close to the end of the rising period, make the gooey butter topping. Heat the oven to 350 degrees. In a small bowl, whisk corn syrup with 2 tablespoons water and the vanilla. Using an electric mixer with paddle attachment, cream 12 T butter, 1 1/2 cups sugar and 1/2 t salt until light and fluffy, 5-7 minutes. Scrape down the sides of the bowl and beat in one egg. Alternately add flour (1 cup plus 3 T total) and corn syrup mixture, scraping down the sides of the bowl between each addition.
Spoon this topping in large dollops over the risen cake and use an offset spatula to gently spread it in an even layer. The cake will be done in 30-45 minutes -- start checking at the 30 mark -- the cake should rise and fall in waves and have a golden brown top, but will be very soft in the center when down. Allow to cool in pan before sprinkling with confectioners' sugar for serving.