Preheat the oven to 375°F (190°C) and grease a round 9-inch pie dish or 9-inch cake pan. (See Notes for muffin pan instructions.) For extra buttery rolls, I like to grease the baking dish with 1–2 teaspoons of melted butter instead of non-stick spray.

Whisk the flour, granulated sugar, baking powder, baking soda, and salt together. Add the egg, buttermilk, and melted butter. Mix with a wooden spoon or rubber spatula until a dough forms. Dough will be very soft and slightly sticky. If it’s too sticky and wouldn’t roll, add more flour, 1 Tablespoon at a time, until dough seems workable. Do not add more flour than you need; a soft dough is good.

Place dough on a floured work surface and, using a lightly floured rolling pin, roll dough into a 10×14-inch (25x35cm) rectangle.

Spread softened butter (the softer it is, the easier it is to spread!) all over the dough. Mix the brown sugar and cinnamon together. Sprinkle all over the top. Roll it up tightly into a 14-inch log. Using a very sharp knife, cut into 12 rolls; each roll is slightly wider than 1 inch. Arrange rolls in the prepared pan.

Bake for 22–26 minutes, until lightly browned around the edges. If you notice the tops are getting too brown too quickly, loosely tent the pan with aluminum foil and continue baking. If you want to be precise about their doneness, their internal temperature taken with an instant read thermometer should be around 195–200°F (91–93°C) when done. Remove pan from the oven and place pan on a wire rack as you make the icing. (You can also make the icing as the rolls bake.)

In a medium bowl using a handheld or stand mixer fitted with a paddle or whisk attachment, beat the cream cheese on high speed until smooth and creamy. Add the butter and beat until smooth and combined, then beat in the confectioners’ sugar and vanilla until combined. Using a knife or icing spatula, spread the icing over the warm rolls and serve immediately.

Cover leftover frosted or unfrosted rolls tightly and store at room temperature for up to 2 days or in the refrigerator for up to 5 days.