To make the dough: Mix together and knead all of the dough ingredients — by hand, mixer, or bread machine — to make a smooth, soft dough.
Place the dough in a lightly oiled bowl, turn to grease all sides, cover the bowl, and let the dough rise until it's nearly doubled in bulk, 1 to 2 hours depending on the warmth of your kitchen.
To fill and shape the buns: Gently deflate the dough, and transfer it to a lightly greased work surface. Roll the dough into a 16x21-inch rectangle.
Spread the dough with the 1/3 cup butter. Mix the brown sugar and cinnamon, and sprinkle it evenly over the dough.
Starting with a short end, roll the dough into a log and cut it into 12 slices.
Place the buns in a lightly greased 9x13-inch pan. Cover the pan and let the buns rise until they're nearly doubled, about 30 minutes.
While the buns are rising, preheat the oven to 400°F.
Uncover the buns, and bake them until they're golden brown, about 15 minutes. While the buns are baking, make the icing.
To make the icing: In a small bowl, beat together the cream cheese, butter, sugar, and vanilla.
Remove the buns from the oven. Spread the icing on the buns while they're warm.
Serve buns warm, or at room temperature. Wrap in plastic and store at room temperature for a day or so; freeze for longer storage.