Place the warm milk and yeast in the bowl of a stand mixer.
Whisk gently and let sit for about 5 minutes.
Add the sugar, salt, butter, egg, vanilla extract, and flour to the bowl.
Use a dough hook attachment to blend the ingredients together at medium speed.
It will take around 10 minutes for the dough to come together and form a smooth ball.
Transfer the dough to a large, greased bowl and cover it with plastic wrap.
Let the dough sit for an hour, allow it to rise, and puff up to about double its original size.
Once the dough has risen, punch it down to deflate, then place it on a lightly floured work surface. Use a rolling pin to roll the dough out to a large rectangle.
Roll up the dough tightly like a Swiss roll.
Cut the roll into 12 equal pieces, slicing slowly to keep the circular shape of each bun as you slice.
Place the slices on a parchment-lined sheet tray, then cover the tray gently with plastic wrap and let it sit for another hour to allow the rolls time to rise again.
Preheat your oven to 350°F / 180°C.
Once the rolls are puffed, remove the plastic wrap and place the tray in the preheated oven.
Bake for 18-25 minutes or until the buns are golden brown.
Remove the buns from the oven and let them cool down.