This coffee cake may take longer to make than others, but the results are worth it. Freeze one for later.
Combine the warm water, milk, 4 tablespoons butter, salt, sugar, cardamom, and eggs in the bowl of an electric mixer fitted with the paddle attachment. Sprinkle yeast over top; let stand until foamy, 10 to 15 minutes.
With the mixer running, gradually add flour. Mix until well combined; dough will be sticky. Turn dough out onto a floured surface, and gently knead until dough forms a ball. Place in a large buttered bowl to rise, covered with plastic wrap, until doubled, about 1 1/2 hours.
Preheat oven to 375 degrees. Butter two 9-inch tube pans; sprinkle bottom of each with 1/4 cup brown sugar. Punch down dough; turn onto lightly floured surface. With a rolling pin, roll into a rectangle that is about 14 by 17 inches and about 1/2 inch thick. Brush a third of the remaining butter over the middle third of the dough, and sprinkle with 1 cup of the filling. Fold the bottom third of dough over the filling and overlap the top third, forming an envelope. Rotate the dough 90 degrees. Roll the dough again into a rectangle roughly the same size, and repeat with butter and half of the remaining filling. Repeat one more time, three rollings in all, using remaining filling. Rotate dough 90 degrees, and let rest, covered with a damp cloth or plastic wrap, for 15 minutes. Roll dough one final time into a long rectangle, 11 by 17 inches and about 1 inch thick. With a sharp knife, cut a 1-inch slice from the end of the dough. Pick it up, twisting it gently into a spiral. Place the twist in the bottom of the pan, forming a circle. Repeat, using remaining dough, filling each tube pan with 2 layers of twisted dough. Cover each cake with plastic wrap, and let rise until doubled, about 45 minutes.
Bake cakes until golden brown, 40 to 50 minutes. Let rest in pans 15 minutes before inverting. Flip cakes over so attractive side is up. Serve cakes warm or at room temperature. After cooling completely, cakes can be frozen, wrapped in two layers of plastic wrap, up to 1 month.