Doable, but not DIY doable unless you can weld/do metal work.

A torsion box would be the basic design, but I think 10' and drawers is beyond a normal wood constructed torsion box.

There would have to be a steel I beam incorporated in the front of the torsion box and supports for each end built into the walls or exposed.

Using good A rated plywood would be much less expensive than veneering the thing. And Formica/etc. laminated to the top/front would be better yet.

Maybe I'm missing something but the image appears to be a countertop. I've seen that done with a piece of angle iron bolted to the wall in the back. The countertop sits on the angle iron.

I'd do the following:
Attach a 2 x 4 attached to the wall to support the back of the desktop.
Make the drawer face frame with full-length 1 x 2 clear hardwood. Glued and doweled together, it would make a strong truss for the front of the desk, (but I still wouldn't recommend sitting on it).
Use two knee braces, located at wall studs, 60 degrees to the wall and coming out to about 6" from the front edge of the desktop, to help keep it from developing a sag.
Use two layers of 3/4" plywood for the desktop (Sounds, looks and feels more substantial than a single layer thickness.)
Regardless of the desk surface material, I'd use real wood for the front edge face. It would hold up better under normal wear and tear.
JMHO
R3

This is real easy. It is a basic 2 pedistal desk. (Pedistal is what I call a stack of drawers that support a desk top)

The pedistals will reduce the center span to 6'. With the continuous ledger along the back wall, you could get away without a center support. When I say you could get away with it, you'll have to build the top out of double thickness Baltic or Russian birch plywood.

My office has this exact setup, the only difference is my unsupported center span is 53". The top is constructed of 3/4" russian birch ply, with the front 3" double thickness. It is laminated with black Wilsonart and has a solid cherry front edge. I weigh about 200 pounds, and when I sit in the middle there is exactly 1/16" deflection. I believe you can span 72" with 1.5" thick solid plywood top, if you build it the same way.

Most people are working with MDF or particle board, so their ideas of span and load capabilities are based on totally different materials.

If the center span proves to be too long, install a countertop cantilever bracket directly to a wall stud near mid span.

The countertop cantilever brackets are designed with the top corner relieved to allow space for the continuous ledger. It is an easy thing install after the fact.

I think it makes more sense to incorporate one or two lengths of steel reinforcements - rectangular section tubing - to resist flexing of the desktop. One at the edge of the counter, concealed by trim; one in the center; the ledger on the back wall supports that edge.

That would be easier than using extremely stout wood construction, 1.5'' thicknesses, etc. No need to weld anything, you can bolt the steel tube to the underside of the counter. Find a metal supply company if the local big box doesn't have anything suitable in 10 foot lengths.

chris,
I think a torsion box is a viable design because it is strong and light. Although 10' is a large span, it will be supported on 3 sides. If you start doubling up 3/4" plywood then you have to consider the weight of the plywood in addition to what ever you use the desk for(computer, books, etc).
As far as the drawers, what do you have in mind? If you are thinking about something like a row of drawers maybe 4-6" high spanning the entire 10', then you wouldn't even need a torsion box. You could mortise and tenon a frame together out of 1x. The rails, stiles, vertical dividers,cross pieces etc would act sort of like a torsion box and you could use 1/2" MDF as the top and a 1/4" or 1/8" sheet of something on the bottom.
If you are thinking about larger file sized drawers hanging from under the bottom of the desk and floating as well, as long as they ran all the way to the back wall and were firmly attached there, it would work.
You definitely want to make some type of template because the walls will not be square.