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Measure yeast and sugar into the bowl of the stand mixer.
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Heat the milk until warm and stir in the 2 tablespoons of butter. It's alright if the butter doesn't melt entirely.
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Pour over the yeast and sugar in the mixing bowl and allow the yeast to foam up.
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Add the egg.
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With the dough hook on add 2 cups of flour and start mixing on low or 2 speed.
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Once mixing add in the last 2 cups of flour with the salt.
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Knead for 4 – 8 minutes on the 2 speed until the dough is a smooth and elastic ball (bounces back when gently indented) and the sides of the mixing bowl are clean.
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Oil a large mixing bowl.
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With clean hands, shape the dough into ball and place into the oiled bowl.
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Cover and allow to rise until doubled in size. 30 – 60 minutes.
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Generously butter a 9″ x 13″ baking pan with 2 teaspoons of softened butter (best done by hand).
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Punch down the dough.
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Roll the dough out on a lightly oiled surface to a 12″ x 15″ rectangle with an even thickness.
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Mix the cinnamon into the cranberry sauce.
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Spread the cranberry sauce evenly on the dough, leaving a 1/2″ border at the far end for sealing.
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Carefully roll the long side of the dough away from you and seal the edge (with a little water from your finger along the edge can help the dough to stick).
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Cut in 12 equal pieces (about an 1″ each) with a floured, serrated knife.
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Place in buttered baking dish.
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Cover and allow to rise for 30 minutes.
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Preheat the oven to 375 degrees.
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Bake for 25 (20-30 depending on oven) minutes, until the rolls are brown and evenly cooked through.
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Remove from the oven and make icing while the buns are still warm.
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Add powdered sugar, cinnamon, vanilla and milk together to make a thick paste. Then slowly continue adding milk or cream, a teaspoon at a time, until reaching desired consistency.
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Slather the thick frosting on the cranberry cinnamon buns while still warm.
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Serve warm or at room temperature.