You might have to search out an older Steelcase highback executive chair, and have it reupolstered if it needs it. They show up on ebay once in a while. The last one I saw was about a year ago. The older designs have a screw type height adjuster, so that you'll never get that sinking feeling as the pneumatic adjuster gives way.
I picked up a set of steelcase chairs back in '86. I still have them. At my heaviest, I was 365. That was in the mid 90's. I'm still 330 pounds, and all the chairs are working just as when I purchased them.
They came in three types; standard office chair with locking tilt, sorta high back (that would be a mid back for you), and the top of the line with a real high back, which I can rest my head on (I'm six foot four) and not reach the top of the chair with the back of my head. That would be your target.
This one would do, but it had a hydraulic height adjuster which is worn out, and of course, the hydraulic or pneumatic adjusters are terrible anyway. But you get the picture of what they look like if you see on for sale. Just watch the height adjustement type. You want the screw type adjuster. These chairs all have an adjustable recliner mechanism, you can adjust the tension as well as lock it in upright position (there's a rocker switch under the left side of the seat).
Good luck. People who own these chairs don't give them up. I had one, and gave it away to a friend who had back problems. That was over ten years ago. He still has it. And I haven't found another one, even though I've been watching online and searching used office furniture stores.
I could sit on that chair for 10 hours and then get up and feel fine. That's how good they are. Steelcase unfortunately had switched to a pneumatic or hydraulic lift mechanism, because that's what 99% of the population of normal sized people want. And it leaves us big people without an adequate office chair.