While the dough is resting, combine the brown sugar and cinnamon in a small bowl.
On a lightly floured surface, roll the dough to an 8-inch square.
Spread the softened butter over the surface of the dough, covering as much as possible. Sprinkle the cinnamon sugar over the butter.
Roll the dough up into an even log. Slice it into eight 1-inch pieces.
If preparing in advance, transfer the rolls to an airtight container, separating each roll with a piece of parchment. Store in the fridge or cooler until you're ready to cook them, up to 2 days.
When you're ready to bake, remove the rolls from the cooler or fridge and arrange them in a single layer in a parchment-lined Dutch oven. Cover loosely, then place in a warm spot and let them rise until puffy and doubled in size, 1 to 1½ hours.
While the rolls are rising, prepare 24 hot coals in a chimney. (Alternatively, you can use hot campfire embers).
When the cinnamon buns have risen, cover the Dutch oven with its lid, and place in your fire pit, with 8 coals directly underneath and 16 coals on the lid.
Bake until the buns are golden brown and cooked through, 35 to 45 minutes.