Danish Cinnamon Snails
1 cup warm milk (110 degrees F/45 degrees C) 2 eggs 2 tablespoons ground cinnamon 2 tablespoons white sugar 3 (0.6 ounce) cakes cake yeast 4 cups all-purpose flour, or as needed 6 ½ tablespoons butter, room temperature ¼ teaspoon salt ½ cup white sugar ½ teaspoon ground cardamom ⅔ cup butter, softened