Make the Dough:

1. Combine the flour, sugar, salt, and yeast in a large bowl; set aside.
2. Combine the milk, water, and butter in a small pot and warm over low heat until the butter is melted and the mixture feels warm to the touch (about 110°F).
3. Add the wet ingredients to the dry, add the egg, and mix until a dough forms.
4. Turn the dough onto a lightly floured surface, knead for four minutes, cover loosely with plastic wrap, and let rest for 10 minutes in a lightly greased bowl.

Make the Filling:

1. Toast the cardamom pods in a small pan over medium-high heat until fragrant, four to five minutes. Transfer to a spice grinder and grind into a powder. Set aside. (Skip this step if using preground cardamom.)
2. Using a stand mixer fitted with a paddle attachment, cream the butter, brown sugar, and ground cardamom until the mixture is unified and creamy, three to four minutes.
3. Once the dough has rested, lightly flour a work surface and roll out to a 9-by-18-inch rectangle. Spread filling in an even layer across the dough to cover completely. Start from one of the short sides of the dough and roll up into a tight log. Cut into 16 even rolls and arrange in a buttered nine-inch round baking pan. Cover with plastic wrap and let rise in a warm place until doubled in size, 90 minutes.
4. Preheat the oven to 375°F. Pour the heavy cream over the top of the buns and put in the oven with a baking sheet underneath to catch any cream that may bubble up and out. Bake until the tops are browned, 20 minutes. (If the buns are getting too brown before time is done, cover with foil and continue baking).

Make the Glaze:

1. While the buns are baking, whisk together the coffee, vanilla, and confectioners' sugar. Remove the buns from the oven, pour the coffee glaze over the buns, and let rest for five minutes. Pull apart and enjoy. (For a pretty presentation, you can flip them out upside down onto a plate.)