Unfortunately, I was not able to find out anything about this interesting-looking house.
Built ca. 1796 for James Powell Cocke, Edgemont is significant as a very early example of a country residence in the combination Palladian and French manner promulgated by Thomas Jefferson. Although the design of the house has been credited to Jefferson for several decades, precise documentation of the authorship remains yet to be established. The character* of the compact and sophisticated dwelling is uniquely Jeffersonian, however, and exhibits the influence he had on the architecture of his region. . . .
— from the 1980 nomination to the National Register of Historic Places
At the time of the photo, only two porticoes of the probable original four still existed. The house was restored and renovated in the late 1930s and 1940s. There are more F.B.J. photos here and 1996 photos here.
* “a formality and classical correctness devoid of monumentality. . .”