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 45 minutes.
Make the sautéed apple filling: In a large nonstick skillet, add the apples, sugar, cinnamon, nutmeg, and cloves and stir until evenly coated. Cook over medium-high heat until the apples begin releasing their juices.
Add the cornstarch to the pan and stir to evenly incorporate. Cook until the apples are tender and coated in a caramel-like sugar and spice mixture. Set aside to cool.
Make the cardamom cinnamon sugar filling: Add the softened butter, dark brown sugar, 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 large skillet or 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 12×20 inch rectangle. Make sure the dough is smooth and evenly thick.
Spread the cinnamon sugar filling all over the dough, then spread the sautéed apples on top of the filling. Tightly roll up the dough to form a 20-inch-long log and cut into 12 even rolls. Arrange them in the prepared pan. Cover the pan and allow the rolls to rise for 20 minutes or until nearly double in size.
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 chai cream cheese glaze: Add the cream cheese and butter to a saucepan over medium-low heat and stir until melted. The mixture will look curdled, don't worry! Add the cinnamon, ginger, cardamom, and allspice and whisk until combined.
Add the powdered sugar and milk to the saucepan and whisk over low heat 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.