To make the dough, put the warm water in a large bowl (or the bowl of your stand mixer) and sprinkle with the yeast and a pinch of the sugar. Let stand 5 minutes, or until the yeast is foamy. (If it doesn’t foam, toss it and buy fresh yeast.)
Add the milk, egg, 2 cups of the flour, butter, salt and remaining sugar to the yeast and stir by hand or with the dough hook attachment of your stand mixer until the dough comes together. Add the rest of the flour until you have a slightly tacky dough. Continue to knead with your dough hook or by hand on the countertop until the dough is smooth and elastic. Cover and let rest for an hour, or until doubled in bulk. (At this point, you could refrigerate the dough overnight and continue assembling the buns the next morning.)
In a small saucepan, combine the butter, brown sugar, maple syrup and water and bring to a simmer, stirring to dissolve the sugar. Pour into the bottom of a pie plate or 9-inch round or square cake pan.
On a lightly floured surface, roll the dough into a 10x15-inch rectangle. Brush with melted butter and scatter with brown sugar; smooth the sugar to evenly distribute it with your hand. Sprinkle with dried cranberries, orange zest and cinnamon.
Starting on a long side, roll the dough up into a log. Using a serrated knife, cut it crosswise into thirds. Cut each piece into thirds – this is easier than eyeballing it to get 9 even pieces. Place the rolls cut-side-up into the pans, placing one in the middle and the rest around it, or in the case of a square pan, in 3 rows of 3. Cover with a tea towel and let rise for another hour, until doubled in bulk. (If you’re making them the night before, cover and place in the fridge for a slow rise; take them out and leave them on the countertop for 1/2 hour or so before baking.)
When you’re ready to bake, preheat the oven to 350?F. Put a baking sheet on the rack underneath (to catch any drips) and bake for 30 minutes, or until deep golden. Let cool for 5-10 minutes, but invert onto a plate while still warm. (If you wait too long and they get stuck in the pan, slide back into a hot oven to rewarm, then try again.)
To make the icing, stir enough orange juice into the sugar to make a drizzling consistency. Drizzle over the buns. Makes 9 cranberry-orange cinnamon buns.