In a small bowl, combine warm milk, sugar, and yeast. Let it sit for 5-10 minutes until foamy.
In a large bowl, whisk together flour and salt. Make a well in the center and pour in the yeast mixture, melted butter, and egg. Mix until a shaggy dough forms.
Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm place for 1 hour, or until doubled in size.
While the dough is rising, prepare the filling: In a small bowl, mix together brown sugar and cinnamon. Softened butter will be used to spread on the dough.
Prepare the caramel topping: In a saucepan over medium heat, melt the butter. Stir in the brown sugar and corn syrup. Bring to a boil, then remove from heat and stir in the vanilla extract. Pour into the bottom of a 9x13 inch baking pan.
Punch down the risen dough and roll it out into a 12x9 inch rectangle on a lightly floured surface.
Spread the softened butter evenly over the dough, leaving a small border. Sprinkle the cinnamon-sugar mixture evenly over the buttered surface.
Starting from the long side, tightly roll up the dough. Cut the roll into 12 equal slices.
Place the sticky bun slices cut-side down into the prepared baking pan over the caramel topping.
Cover loosely with plastic wrap and let rise for another 20-30 minutes.
Preheat oven to 350°F (175°C).
Bake for 20-25 minutes, or until golden brown and bubbly.
Let cool in the pan for 10 minutes, then invert onto a serving platter. Serve warm.
Notes
For extra gooeyness, you can add chopped pecans or walnuts to the filling before rolling.