Make the dough
Combine the flour, sugar, salt, and yeast in a large bowl; set aside. 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). Add the wet ingredients to the dry ingredients, add the egg, and mix until a dough forms. On a lightly floured surface, knead the dough for 4 minutes, then cover loosely with plastic wrap and let rest for 10 minutes in a lightly greased bowl.
Make the filling
Toast the cardamom pods in a small pan over medium-high heat until fragrant, 4 to 5 minutes. Transfer to a spice grinder and grind into a powder; set aside (skip this step if using pre-ground cardamom). Using a stand mixer fitted with a paddle attachment, cream the butter, brown sugar, and ground cardamom until unified and creamy, 3 to 4 minutes.
Shape and bake the buns
Once the dough has rested, lightly flour a work surface and roll it out to a 9 × 18-inch rectangle. Spread the filling evenly over the dough to cover completely. Starting from one of the short sides, roll the dough into a tight log. Cut into 16 even rolls and arrange in a buttered 9-inch round baking pan. Cover with plastic wrap and let rise in a warm place until doubled in size, about 90 minutes. Preheat the oven to 375°F. Pour the heavy cream over the buns and place the pan in the oven with a baking sheet underneath to catch any cream that may bubble over. Bake until the tops are browned, about 20 minutes (if the buns brown too quickly, cover with foil and continue baking).
Make the glaze
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 top, and let rest for 5 minutes. Pull apart and enjoy. (For a pretty presentation, you can flip them out upside down onto a plate.)
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