In a small bowl, combine the warm water, pinch of sugar, and the yeast. Set aside to get all bubbly while you measure out the rest of the ingredients (about 10 minutes).
Place the flour, sugar, and fine kosher salt in a stand mixer fitted with the paddle attachment. Stir them together for a few seconds on low just to get everything evenly mixed.
Meanwhile combine the milk with the butter in a microwave-safe bowl and heat it in the microwave for a minute or so, just enough to melt the butter. Let stand at room temperature to cool slightly.
When all the components are ready, add the milk and butter mixture to the mixer, along with the foamy yeast mixture and the two eggs. Stir them together on low speed to form a shaggy, evenly mixed dough.
Switch to the hook attachment to begin kneading on low speed. If the dough seems too wet, add up to 30 grams (¼ cup) of all purpose flour. The dough should not be so sticky and it shouldn't stick to the bowl. It should feel tacky.
Knead the dough for 8 minutes, then dump it out on the counter and knead it by hand for a minute or so to form a smooth round ball of dough.
Oil a medium bowl and place the smooth ball of dough in it. Cover with plastic wrap and let it rise for 1 hour in a warm place or until doubled. Prepare the filling while you wait.
Combine the melted butter with the brown sugar and cinnamon to form a thick paste (it will thicken as it rests and cools to room temperature, especially if the butter was very warm).
When the dough has doubled, remove it from the oven (or fridge, in this case you should let it warm up to room temperature!) and punch it down.
Roll the dough on a lightly floured surface to 9x15 (don't worry too much if it's bigger).
Spread the filling over the entire surface of the dough, then roll tightly into a 15 inch log.
Cut the dough into 12 equal buns (you don't have to trim the ends) and place in a butter 9-inch round cake pan.
Cover with a damp towel and let them rise again for about 30 minutes.
Preheat the oven to 350°F. Bake the rolls for 25 to 30 minutes. Remove the pan from the oven and place the pan on a wire rack to cool.
Prepare the icing by whisking together the cream cheese, powdered sugar, milk, and rum. If your icing is too thin, add a touch more powdered sugar. If your icing is too thick, dilute it with a little milk.
Serve the cinnamon buns with a dollop of icing and enjoy!