Prepare your brown butter. Melt the butter in a pan over medium heat. Ideally, use a light-colored pan so you can see the butter browning. Stir the butter constantly. The butter will foam and sizzle and gradually turn golden brown, with a strong nutty smell. Immediately remove from heat and pour into a heat-proof container. Let cool before using it in the dough.
In the bowl of a stand mixer fitted with a dough hook attachment, mix the yeast, sugar, and lukewarm milk. Let it sit for 5-10 minutes until the yeast foams. If there are no signs of yeast activity, it could mean that your yeast is dead. Throw it out and try again.
Add the brown butter, flour, egg, pumpkin puree, salt, dried rosemary, and cardamom into the bowl. Knead the dough on medium-low speed for 20 - 25 minutes. Your stand mixer should be going at a steady speed, but should not be going so quickly that the dough is being slung around. At first, the dough will be very wet and slack. Over time, it will tighten up and should pull away completely from the sides of the bowl. Periodically check to see if the dough passes the window pane test. Cover the bowl and let the dough proof for 45 minutes to 1 hour until doubled in size.
Grease your baking tin. Remove the dough from the bowl onto a lightly floured surface. Divide into 8 equal pieces and shape each piece into a smooth, round ball. Arrange the rolls evenly within the baking tin.
Cover the tin and let proof for another 45 minutes to 1 hour. When you press your finger into the dough, it should bounce back halfway. Towards the end of the proof, preheat the oven to 375°F.
Beat together the egg and milk to create an egg wash. Brush the tops of the rolls and bake for 20 - 25 minutes until golden brown and makes a hollow sound when tapped.
Let cool in the tin for a few minutes before removing and serving.