I THINK I KILLED THE 340

I have run the Q on my 367 as tight as .028 without problems, but have been running .032 since 2004. And
With alloy heads, she has run up to 195psi, still on 87 gas.
So, here are my guesses:

1) Or
you could just mill some shallow bowls in your alloy heads, one above each chamber.
Adding .032Q, and the gasket .039, then subtracting .034pop-up, I get .037chambers. You could share that with the piston crowns, depending on your piston to valve clearance.

2) Or
just put the X-heads back on.
As I recall, the chambers on those are around .100 deep; to which you add the .039gasket, and then subtract the .034 pop up, and I get .104 quench which in terms of Detonation resistance, is good to go.

3) Or
IIRC, there is available, an .052 composition gasket. Subtracting .034 from that, I get a crown-cut of just .014 to get a Q of .032.

4) By selecting the right cam,
you may not have to deepen the eyebrows, or if you do then not by much, and maybe only the intakes. You can buy a tool for that and do it in your garage.

5) In any case,
your probably gonna end up with a lotta SCR, which, if using the iron heads, is gonna take a late closing intake event, to get the pressure down to a safe level.

6) So then
I'm favoring the alloy heads on this one. If you haven't already bought some, Edelbrock makes 'em just for the 340, with 65cc chambers, which by the math makes the depth of the chambers about .020; tho I haven't measured them..
So then, with the thick gasket, you have;
.052 + .020, less the .034 pop-up; I get a Q of .038, dang near perfect.

7) piston to valve clearance is gonna have to be measured regardless.

8) but above all, check the distance from the top ring to the deck. If the compression ring gets to be too close to the action, yur gonna have trouble, there is no water up there. So then,

9) you may need some shorter rods..