For the dough: Coat a large bowl with butter and set aside. Place milk and water in a heatproof bowl or measuring cup and warm in the microwave until the mixture reaches 100°F to 115°F (this can also be done in a small saucepan on the stovetop over low heat). Once milk mixture is warm, transfer to the bowl of a stand mixer and sprinkle yeast on top. Set aside to rest until mixture bubbles, about 5 to 10 minutes. (If the mixture does not bubble, either the milk mixture was not at the correct temperature or the yeast was old.) Add eggs, sugar, and salt and whisk until evenly combined. Add flour and place on a stand mixer fitted with a dough hook. Mix on low speed until flour is mixed in and dough just comes together, about 1 minute. Increase speed to medium high and add butter piece by piece, letting each piece completely incorporate before adding the next. Continue mixing until butter is completely incorporated and dough is smooth and pulling away from the sides (the dough will be soft, moist, and slightly sticky), about 10 minutes total. Transfer dough to the prepared bowl and turn to coat in butter. Cover the bowl with plastic wrap and leave in a warm place until dough doubles in size, about 1 1/2 hours. Punch dough down, cover, return to the warm place, and let rise again until doubled, about 45 to 60 minutes. For the topping: Coat two 8-inch cake pans with butter; set aside. Melt 4 tablespoons butter in a medium saucepan over medium heat; stir in sugar and milk. Continue to cook, stirring occasionally, until boiling, about 10 minutes. Divide caramel evenly between the cake pans; set aside. For the filling: In a small bowl, combine sugar and cardamom and mix well; set aside. To assemble: On a lightly floured work surface with a lightly floured rolling pin, roll dough into a 14-by-14-inch square. Using your hands, spread 3 tablespoons room-temperature butter evenly over dough, then sprinkle with filling. Starting at the edge closest to you, roll dough into a cylindrical shape. As necessary, stretch dough to ensure that it is being rolled evenly. Using a sharp knife, slice dough cylinder into 12 pieces (they will resemble slices of a tortilla wrap). Place 1 roll in the center of each cake pan. Evenly space 5 rolls around each center roll. (At this point you can cover the pans with plastic wrap and refrigerate them overnight.) If baking immediately, set, uncovered, in a warm place to rise for 40 minutes. Heat the oven to 350°F and arrange a rack in the middle. If the rolls have been refrigerated, take them out of the refrigerator and let sit at room temperature for 40 minutes before baking. Bake the rolls until they’re puffed, golden brown, and an instant-read thermometer inserted into the center roll reads 190°F, about 35 minutes. Remove from the oven and immediately invert the pans onto a serving plate. Let the rolls sit in the inverted pans 1 to 2 minutes before removing the pans.