There is an article in the Sept/Oct 2000 Issue of Fine Woodworking (Issue 144) by Lon Schleining on building the desk. It uses modern techniques to avoid some of the problems with the original that resulted in some serious warping of the writing surfaces and damage to their hinges.
From time to time, Mario Rodriquez teaches a class on construction of the desk at the Philadelphia Furniture Workshop.
A lovely project. I've done four of these, but went to solid, well quartered mahogany for everything, versus the laminated construction shown...none have really moved at all, due to well seasoned Cuban or SA mahogany stock. Three of the four were gifts to outgoing battalion or brigade commanders (all Army officers develop an interest in Revolutionary and Civil War history as a job skill, and spend a lot of time in the woods, so appropriate), and one to a retiring Naval officer...all prized by their owners.
If you can source it, or resaw it from thick boards and edge glue it up into wide panels, I agree that QS mahogany is the best way to solve the warping problem.
Note that Mario Rodriguez' interpretation solves the problem differently, by using breadboard ends on the writing boards. he also used a different kind of hinge. Both compromises are clearly visible if you know what to look for, and detract from the historical accuracy of the reproduction.
Unless you are planning to keep it in a glass case in a museum, like the original, something needs to be done to prevent the writing boards from cupping and wrecking the hinges.