Fit a stand mixer with a dough hook. Combine starter and milk until incorporated. Add sugar, eggs and salt to combine. On low, start adding flour cup by cup. Your ratio will depend on how wet or dry your starter is. When you’ve reached three cups, add more gradually until dough just starts to come away from the bowl. Turn mixer up to speed 2 until dough forms a ball and isn’t sticky. Turn out onto a floured surface and knead until smooth and elastic.
Place dough into an oiled bowl and cover with plastic wrap or a damp cloth. Let rise in a warm spot until doubled in size, approximately 2-3 hours. This will depend on how warm your kitchen is.
Once your dough has doubled, punch it down and fold it back into a ball to rise again. After the second rise, punch down once more and roll out with a floured rolling pin to a rectangle.
Spread butter generously over your dough, leaving a 1-inch border on your bottom lengthwise end. Mix sugar and cinnamon together and spread evenly over the butter. Roll from the buttered edge inwards towards the un-buttered end. Roll fairly tight,but not too tight as the middles will pop out if they don’t have enough room to expand during baking. Once you’ve reached the butter-less end, dab some water along the naked dough to secure it.
Cut log into 1 inch rounds. Grease and parchment a 9-inch pie pan and place your rolls inside. Dental floss also works well - thread floss underneath and use a tying motion to cut rolls without squishing.Bake at 400 Fahrenheit for 10-15 minutes until set and golden.
Mix all icing ingredients together and pour over buns when they've cooled.