Make the tangzhong. Combine the tangzhong ingredients (¼ cup flour and ⅔ cup milk) in a small saucepan and whisk until no lumps remain.
Place the saucepan over medium heat and cook, whisking constantly, until the mixture thickens into the consistency of smooth mashed potatoes, around 2 minutes.
Remove saucepan from heat and scrape the tangzhong into a small bowl, then cover with plastic wrap and let cool slightly. To speed up this process, you can refrigerate the bowl and occasionally stir until it's room temperature or slightly warm.
Prepare the cinnamon roll dough. Whisk together the all-purpose flour, instant yeast, light brown sugar, and salt in the bowl of a stand mixer.
Make a well in the center of the dry ingredients, then add in the cooled tangzhong, pumpkin purée, milk, egg, egg yolk, and partially melted butter.
Transfer the bowl to a stand mixer fitted with the dough hook attachment and knead on medium-low speed for around 10 minutes. The dough should pull away from the sides into a smooth ball and be slightly sticky.
Once everything comes together, continue to mix until the dough passes the windowpane test. To do this, pinch off a golf-ball sized piece of dough and stretch it between your fingers; if it's mixed enough, you will be able to stretch it thin enough for light to pass through without the dough breaking. If the dough tears, continue to knead for 5 minutes, then repeat the test. If the dough becomes too sticky, add more all-purpose flour as needed, one tablespoon at a time (try not to add too much flour, otherwise the rolls will become too dry).
Proof. Shape the dough into a smooth round and place into a lightly greased bowl. Cover with a damp towel or plastic wrap and let it rise at room temperature until doubled in size, around 1-2 hours (this may happen slower or faster, depending on how warm your kitchen is). If needed, you may also proof the dough in the fridge overnight.
To check if the dough is done rising, poke it with one finger; if the dough springs back and leaves a slight indentation, it's ready to be shaped. If it springs back with no indentation, it needs longer. If it doesn't spring back at all, it's overproofed.
While the dough rises, make the pumpkin filling1. Add the 1 ½ tablespoons pumpkin spice and dash of cardamom (if using) to a medium heat-safe bowl and set aside.
Place the ½ cup cubed butter into a large saucepan and set over medium heat, stirring until melted. When the butter begins to sputter, stir every five seconds or so until it bubbles and begins to foam. Once you see brown bits floating in the butter and a nutty aroma is released, immediately remove the pan from the heat.
To stop the cooking process, pour the butter into the mixing bowl with the spices, making sure to scrape all the browned bits from the bottom of the pan. Stir the butter and spices together until well incorporated, then set in the fridge to cool completely. To speed up this process, you can set the bowl of brown butter over an ice bath and whisk constantly until it reaches a softened butter consistency, around 5 minutes.
Once the brown butter has resolidified, add in the brown sugar, pumpkin puree, salt, and vanilla, then beat with an electric mixer until smooth. Whisk in the heavy cream, one tablespoon at a time, until a spreadable paste consistency is reached.
Shape the rolls. Lightly grease either two metal 9-inch round pans, two 9x9 inch square pans, or one metal 9x13 pan with butter and parchment paper. Set aside.
Once the dough has doubled in size, gently turn it onto a floured work surface and roll into an approximate 14x19" rectangle. Evenly spread the prepared pumpkin filling over the dough, leaving a ¼ inch margin around the long edge.
Starting from the longer 19'' side, roll the dough into a tight spiral and rest seam side down, carefully pinching the edges to seal. Using floss or a sharp knife, trim off an inch of dough from each end. Slice the remaining roll into 15-20 even pieces, each around 1 inch in width.
Arrange the rolls into the prepared pans, leaving around 1 inch of space between each roll. Cover with a damp towel or plastic wrap and leave to rise for 30 to 60 minutes, or until puffed and almost doubled in size—if you poke the dough, it should spring back slightly and leave a small indentation.
Bake. In the meantime, preheat the oven to 350℉. Once proofed, bake the cinnamon rolls for 15-25 minutes, or until just slightly golden on top and the internal temperature registers 190℉. Do not overbake—the rolls will not take on much color, and overcooking them will make them dry. Allow the pan to cool for at least 10 minutes.
In the meantime, prepare the frosting. In a medium-sized bowl, beat together the cream cheese and butter with a hand mixer until smooth. Add in the pumpkin pie spice, vanilla, salt, and powdered sugar, then mix until an icing forms. Spread the frosting over the cinnamon rolls and serve warm.