Warm milk in microwave for about 30 to 45 seconds. You just want it to be warm, make sure it doesn't exceed 120F, as this will kill the yeast.
Mix yeast and 1 T sugar, add yeast/sugar mixture to warm milk, stir to combine and set aside for five minutes (the yeast will be bubbling, that shows that it's active)
Using paddle attachment on mixer, cream the butter, remaining sugar, and salt.
Scrap down sides of bowl, add egg, mix on medium speed until smooth.
Switch mixer to low speed, add the yeast mixture and flour gradually. Once incorporated, increase to medium speed and beat until it forms a smooth mass and pulls away from the side of the bowl (approx.
to 10 minutes).
Remove dough from ball and on a lightly floured surface knead for one minute.
Place dough ball into a lightly greased bowl, cover with lint resistant towel, or plastic wrap, until it doubles in size (approximately 1 hour)
While dough is fermenting, prepare topping.
On medium/low speed blend cream cheese until smooth. Add eggs, one at a time, until incorporated, scrap down sides as needed.
Pour in one stick of melted butter (it's not called gooey BUTTER cake for nothing), mix until incorporated into cream cheese/egg mixture Turn mixer down to low speed, gradually add in confectioner's sugar, blend until smooth.
Add vanilla, increase to medium speed, blend for five minutes until mixture is light and fluffy.
Lightly grease a 9x13 inch glass baking pan. Press dough evenly into the pan, form a slight edge to prevent "goo" from escaping.
Dock the dough so any steam can escape and not puff up the dough.
Pour buttery cream cheese mixture onto the dough, spread and bake @ 350F for 35 to 45 minutes.
The middle will be liquidy/giggly but the base should be golden brown when done.
Cool in pan, serve at room temperature.