St Louis Gooey Butter Cake

Ingredients

• 3 tablespoons milk at room temperature
• 1 3/4 teaspoons active dry yeast
• 6 tablespoons unsalted butter at room temperature
• 3 tablespoons sugar
• 1 teaspoon kosher salt
• 1 large egg
• 1 3/4 cups all-purpose flour
• 3 tablespoons plus 1 teaspoon light corn syrup
• 2 1/2 teaspoons vanilla extract
• 12 tablespoons (1 1/2 sticks) unsalted butter, at room temperature
• 1 1/2 cups sugar
• 1/2 teaspoon kosher salt
• 1 large egg
• 1 cup plus 3 tablespoons all-purpose flour