Mix together and knead all of the dough ingredients in a stand mixer. Let the mixer knead the dough for about 2 minutes.
The dough should be soft and smooth.
Place the dough in a lightly oiled bowl, turn to grease all sides, cover the bowl, and let the dough rise until it's nearly doubled in bulk, 1 to 2 hours depending on the warmth of your kitchen.
Just before rising time is over, put the almond paste and 1 egg in a food processor. Process it until you have a nice paste.
To fill and shape the buns: Gently deflate the dough, and transfer it to a lightly greased work surface. Roll the dough into a 16" x 21" (40.64 cm x 53.34 cm) rectangle.
Spread the dough with the 1/3 cup butter. Mix the powdered sugar and cocoa and sprinkle it evenly over the dough.
Cover that with the almond paste mixture. Spread it all the way to the edges.
Starting with a short end, roll the dough into a log and cut it into 12 slices.
Place the buns in a lightly greased 9" x 13" (33 x 23 x 5 cm) pan. Cover the pan and let the buns rise until they are nearly doubled about 30 minutes.
While the buns are rising, preheat the oven to 400 °F (204 °C).
When the buns have risen, bake them until they are golden brown, about 15 minutes.
While the buns are baking, make the icing. In a small bowl, beat together the cream cheese, butter, sugar, and rum flavoring.
Remove the buns from the oven. Spread the icing on the buns while they are warm.
Serve the buns warm, or at room temperature. Wrap in plastic and store at room temperature for a day or so; freeze for longer storage.
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