For the cake:
Place oven rack to middle position and heat oven to 350°F.
Grease 13 by 9-inch baking dish.
In bowl of stand mixer fitted with dough hook, combine flour, milk, sugar, egg, yeast, and salt.
Knead on low speed until dough comes together, about 2 minutes.
With the mixer running, add butter 1 piece at a time.
Increase speed to medium-high and continue to knead until dough forms stretchy, web-like strands on sides of bowl, about 6 minutes longer (dough will be soft and sticky).
Using greased rubber spatula, transfer dough to prepared dish.
Using your floured hands, press dough into even layer to edges of dish.
Cover dish tightly with plastic wrap and let dough rise at room temperature until slightly puffy, about 1 hour.