The oldest house in Missouri is the French Colonial style 1792 Louis Bolduc House in Sainte Genevieve. Essentially a timber frame with mud-straw-hair infill, Norman roof trusses and a shake roof. Photos from present day and from 1936 before a restitution.
Not a complicated build today and all construction plans are available at the Library of Congress.
There's a few contenders for the "oldest house in Missouri"-title, all in Ste. Genevieve, like the Green Tree Tavern, but it has been moved and the building date is a little unclear. It could be older.
I don't think anyone specifically teaches how to build this kind of French colonial in the U.S. today but here is a nice looking timber framing school in North Carolina, where you can learn enough to have a go at any kind of timber framing: wildabundance.net
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