Make the vanilla bean brioche dough: In the bowl of a stand mixer or in a large mixing bowl, mix together the yeast with the milk. Add the eggs, butter, flour, salt, sugar, and vanilla bean paste and mix until the ingredients are just barely combined. Let the mixture stand for 5 minutes so the flour has time to hydrate.
Using the hook attachment, beat on medium speed until a soft dough forms. Increase speed to medium-high and beat until the dough is soft and supple, about 6 minutes longer. *If you do not have a stand-mixer with a hook attachment, knead the dough by hand in this step.*
Transfer the dough to a lightly floured surface and knead it with your hands for 1 minute. Form the dough into a ball and transfer it to a lightly greased bowl. Cover the dough loosely with plastic wrap, a paper towel, or aluminum foil and let sit in a warm place until doubled in size, about 30 to 45 minutes.
Make the cardamom cinnamon sugar filling: While the rolls are rising, make the filling. Add the softened butter, dark brown sugar, cardamom, and cinnamon to a medium bowl and combine using a fork until a thick paste forms. Set aside.
Grease the bottom and sides of a 9×13 inch baking dish. Turn the risen dough out onto a lightly floured work surface, punch it down and, using a rolling pin, roll into a 12×18 inch rectangle. Make sure the dough is smooth and evenly thick.
Spread the cardamom cinnamon sugar filling all over the dough. Tightly roll up the dough to form an 18-inch-long log and cut into 12 even rolls. Arrange them in the prepared baking pan. Cover the pan and allow the rolls to rise for 20 minutes or until nearly double.
Preheat oven to 375°F (191°C).
Once the rolls have risen, pour the heavy cream over the top of the rolls, allowing it to soak down in and around the rolls. (The cream shouldn't be hot to the touch, just slightly warm.)
Bake the rolls for 18-20 minutes, until the rolls are lightly golden brown and the center rolls are cooked through. Remove pan from the oven and allow to cool on a wire rack for about 10 minutes as you make the icing.
Make the brown butter vanilla bean cream cheese icing: Cube the butter and place in a light-colored skillet. (Light colored helps you determine when the butter begins browning.) Melt the butter over medium heat, stirring constantly. Once melted, the butter will start to foam. Keep stirring. After 5-8 minutes, the butter will begin browning; you'll notice lightly browned specks begin to form at the bottom of the pan, and you'll start to smell a nutty aroma. Once browned, immediately remove from heat, pour into a medium heatproof mixing bowl.
Add the softened cream cheese to the butter mixture and whisk to combine. Add the remaining icing ingredients to the bowl and whisk until smooth and combined. If the icing is too thin, add a little more powdered sugar; if it's too thick, add more milk. Drizzle icing over warm rolls. Cover leftover frosted or unfrosted rolls tightly and store at room temperature for up to 3 days.