FWIW, Rainier, Shasta, Hood and Whitney really don't have a counterpart in New Hampshire. Those mountains in spring/summer - which I assume is when you've climbed them (feel free to correct me if I'm wrong; also, route choice? Liberty Ridge is pretty different from DC, etc.) - have almost nothing in common with the Whites in Winter. The requisite skills are really different, as are the safety considerations, so it's hard to think of an analagous itinerary. Here are some thoughts, which you can work into your planning as you see fit:
1. The Presidential Range (the ridge from Madison through to Pierce) offers the largest expanse of above treeline terrain in the Whites. Second most would be Lafayette through Flume (AKA Franconia Ridge). Then there are the Carters, the Bonds, and a couple of other less prominent ridges. Baxter State Park in Maine offers similar (though colder and more remote) terrain, but sounds like you're most interested in the Whites. So take your pick from the first two in the list, probably.
2. December is a difficult time to hike in the Whites - IMO, the most difficult. Days are really short, snow pack is often shallow (though sometimes epic; you just never know), and it's often FREEZING cold - well below zero. Though sometimes it's raining and 40 degrees. Or you could be lucky enough to catch an ice storm. So fun. Very difficult/impossible to know what conditions will be like ahead of time.
3. I'm sure a single day Presi-traverse has been done in December, but it's VERY uncommon. Again, very short days and often very difficult conditions - verglas over talus is probably most common, there's usually poor visibility, and blowing snow and freezing fog with wind speeds in excess of 50 mph (often in excess of 100 mph) are common. Best to be very good at navigating in the absence of any terrain features while the wind roars in your ears.
4. None of this is meant to deter you, you can make your own decisions, I'm just trying to outline some of the differences between the mountains out west that you described and the challenges they present (altitude, glacier travel, avalanches, steep snow climbs) and what you might expect in the Whites in December.
With that out of the way, here are some possibilities. I assume you want a single overnight, or possible two nights out? Camping, or hut?
1. Good weather option: Northern Presi traverse. Start at Appalacia, Valley way up to Madison Spring hut (closed during winter), up and down Madison, then over Adams and down to Gray Knob Cabin (see details by searching online). Day two, up and over Jefferson and Washington, then down either to the Cog railway parking area or via Lion Head route (Winter Route probably won't be open yet, so you'll need to navigate steep, icy talus) to Pinkham. This is a big itinerary, very difficult for most people, and dicy at best in questionable weather.
2. Franconia traverse: start at Lincoln Woods trailhead, Osseo trail to Flume then to Liberty, camp at Liberty Spring tentsite. Day two, up Lincoln then Lafayette, decend Old Bridal Path. Franconia Ridge is above treeline, but not quite as nasty as the Northern Presidentials.
3. Full Pemi loop. This is usually a 4-day itinerary: Day 1 same as #2 above. Day two, instead of descending at Lafayette, continue on to Mt. Garfield, camp at Garfield Ridge tentsite (has a nice 3-walled shelter). Day 3 Climb to South Twin then on to Guyot shelter and tentsite. Day 4 up and over the Bonds and back to where you came. Note this is a big itinerary but manageable and not quite so weather dependent since you're not above treeline as much.
Again, I want to emphasize that the Whites in December are very unpredictable. You'll probably get a lot of notes of caution in this thread - take them seriously. If you want to provide more details on your objectives and skills we (collectively) can hone an itinerary.
Cheers,
Brian
