wonder if anyone has taken two identical new phones, left one unused, and used the other for X number of hours, then compared the two. Isn't this the only way to find out if the break in is a real phenomenon and what kind it is?
That doesn't work.
It is a false assumption. There are production variations between identical headphone models. Manufacturers have a range of variation that passes QC. (You'll also find this in almost all electronic parts; resistors and capacitors typically vary +/-10%, so a 100 Ohm resistor might measure between 90-110 Ohms.) Instead of rejecting all drivers that aren't spot-on, they match two closely so they'll be balanced for listening. So one pair might have two drivers matched nicely at one value and another pair matched nicely at a different value. These two headphones will sound different from each other no matter if one is burned-in and the other isn't.
Take a look at the HD-800. I don't know if Sennheiser still offers charts, but they used to email buyers a graph of driver responses for each pair. They really do vary. If the HD-800 varies, you better believe that every other pair does, too. Nothing is as precise as you might think.
And referring to resistors and caps earlier, you better believe that left/right in your amps and sources aren't as precisely matched as you think. They're probably between 5% and 10% off. I hand-matched components in DIY speaker crossovers and I had to buy a bunch of extra parts and spent a few hours measuring and labelling them. I got left and right mirrored to 0.005, but you only get that kind of matching if you DIY or pay for a custom build. Typical audiophile gear won't be matched as closely.
Anyhow, I run every piece of gear right out of the box and have never bothered with burn-in. Everything sounds fine with zero hours on it.
If you don't like the sound signature, don't expect big changes. Not every headphone is for everyone. If you don't like it, you have a few options.
First, either return it or sell it here. Put the money towards headphones you actually like. Unless you've spent a lot of time listening to different models and attended a few meets, you probably don't know what you like yet. Which is OK. This is a learning experience and most people go through a bunch of headphones before they settle in with a pair. If you didn't find the right pair at first, don't worry. That happens to everyone. Almost no one settles down with their first pair. It means you're just like everyone else.
You csn get a slight variation with amplification. If a headphone is almost there, you might be able to get there with a different amp. But don't count on it. It's better to find a headphone agreeable to you.
As a last resort, you can use EQ to change a headphone. But EQ introduces other complications. It's better to find a headphone you like.
But don't count on burn-in to turn something you dislike into something you do like. That's like trusting palm reading or tarot cards for major life decisions. Lots of hokum and very little in the real world.