No such thing. $405 will get you a really nice setup, but it is hardly a 'regardless of price' budget. People spend thousands in headphone gear alone...
As far as headphones, Beyerdynamic and Sennheiser are hard to go wrong with. The Sennheiser HD650's, and Beyerdynamic DT880/990's all fit within your budget, though the Sennheiser's may be a little tougher to find within your price range.
Now, I hope you have something to power those headphones, because onboard audio is NOT going to cut it. Not only is it going to give poor audio quality, but onboard simply will not have enough power to drive them. Logan just did a video on some headphone basics, but to put it simply enough, you need a good headphone amp to drive cans like those, and if you don't have one, then you would be better off getting a slightly cheaper set of headphones, and spending money on a good headphone amp/DAC.
The one Logan recommended, the JDS Lab's ODAC O2 is good, FiiO E09k works awesome, Schiit Audio makes good stuff, ASUS makes some soundcards that will drive them (even though Logan spoke against this idea, I can at least vouch for the performance of the ASUS cards).
So, with that in mind, do you already have a headphone amp to drive a pair of high end cans, or do you need one as well. This would put your budget more in the $200-300 range for headphones, leaving $100-200 for a good DAC/amp.
Missed the part where you said headset, not headphones. If you are dead-set on a headset, the Sennheiser PC363D is not horrible. The headphones themselves are actually of fairly good quality, pretty good mic on it, and the inline DAC/headphone amp will fit most bills. Considering what you are coming from, it will blow those Logitech sets out of the water, and do better than most any other headset.
However, if you are willing to jump outside the headset box, then there are plenty of other options. As mentioned, you can just buy mic to suite your needs, and then pick a pair of headphones to suite what you listen to.
The question is what are your listening habits? Music types?
It's everything from Black/death Metal, Classical, Trance, house, dubstep and punk. But I use a lot more time gaming than listening to music so something that will give a clear, detailed sound and pref to have something that can give a good "surround" sound :)
Definitely check out the Sennheiser PC363D. The bass won't be nearly as punchy as something like the V-Modas when it comes to EDM, but it will put the V-Modas to shame on how much more detail you will gain. If possible, try to give the HD558's, HD555's, or PC360's a test listen to see if you like the sound first, but my vote goes to the PC363D for you.
Real headphones are better. Sennheiser, Beyerdynamic, and Audiotechnica all have a version of their best selling headphones with a mic slapped on for a premium - get one of those, not a headset from a peripheral company.