Hinging the leaf of a drop front desk used to be a simple matter of selecting an appropriately proportioned desk hinge and mortising them into the writing surface of the desk. Desk hinges are nothing more than extra-wide butt hinges.
Desk hinges are mortised into the writing surface with the hinge barrel projecting up and it is with this feature that today's furniture maker takes issue. The argument against this 300 year old solution centers on disbelief that another hinge does not exist that can be flush mounted with no upward projection and still perform this same simple function. The hope is that a drop leaf hinge or other hinge with holes countersunk on the reverse sides will allow the hinge barrel to be imbedded beneath the surface. Even if the hinge is able to pivot backwards 90°, this cannot work without a huge gap between desk and flap. Otherwise the upper edges of the flap and desk will bind against each other long before the flap closes (see below).
The flap binds both because it pivots below the surface of the desk and because the hinge is folding backwards
After this idea is dismissed the counter hinge seems to offer hope. Counter hinges have no upward projection and employ a link type joint between the two leaves. This type of hinge is seen hinging the access flap of bar and bank counters and naturally seem the obvious choice in this application. The problem with these hinges is their lack of predictable action. With their two linked pivots they allow the writing surface to rise or fall and move in and out just as though connected by a short length of chain. This, and the fact that the rather bulky links will be visible under the flap when closed disqualifies this type of hinge as being a candidate for the job.
The links are clearly visible when the flap is closed
No other hinge suggests itself, ("Soss" type invisible hinges will function but will be fully visible when the flap is closed thus defeating at least part of the objective) so we are left to reconsider the traditional desk hinge. It's only failing is its exposed barrel and this is really not such a problem. The barrel is most of the way back towards the rear of the writing surface and in any event only projects upward about 1/8". 10 generations of letter writers have learned to work around this minor obstruction without serious impact on their output. We can perhaps do the same.