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1
Mix 1 cup flour, starter, milk, white sugar, and yeast together in the bowl of a stand mixer. Let stand until spongy and bubbling, about 5 minutes. Add 1/2 cup butter, egg, and salt; stir to combine. Add remaining flour 1 cup at at time until a soft, slightly sticky dough forms.
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2
Transfer dough to a floured work surface; knead until dough comes together, 3 to 5 minutes. Place in a greased bowl. Cover with plastic wrap; allow to rest until doubled in size, about 1 hour.
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3
Preheat oven to 375 degrees F (190 degrees C). Grease a 9x13-inch baking pan.
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4
Punch dough down; transfer back to the floured surface. Roll into a square 1/4-inch to 1/3-inch thick. Spread 1/2 cup butter on top; sprinkle brown sugar and cinnamon on top. Roll dough up tightly; pinch seam together.
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5
Trim the edges and cut the log into 8 equal pieces using a sharp knife. Place rolls in the prepared baking pan. Cover with plastic wrap; let rise until not quite doubled in size, about 45 minutes.
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6
Bake in the preheated oven until golden brown, about 35 minutes. Remove from oven; cool for 20 minutes.
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7
Mix confectioners' sugar, cream cheese, 1/4 cup butter, and vanilla extract together until frosting is smooth and fluffy. Top each cooled roll with an equal amount of frosting.
