Most adventurous eating in Santa Fe
Epazote has an interesting menu, all kinds of mole's, and even a dish featuring huitlacoche (a corn fungus that's a Mexican delicacy and much more delicious than it sounds).
Los Potrillos is supposed to have really good menudo, and Guadalajara Grill has menudo, birria and roasted kid, if you really want to pull out all the carnivorous stops.
Tune-Up Cafe serves pupusa and a few other Salvadoran specialties along with more common Southwestern-style dishes.
1947 Cerrillos Rd, Santa Fe, NM 87505
There's not a lot of exotic stuff on high end menus in Santa Fe. In addition to Deb's menudo suggestion, a few things that kinda sorta come close:
I see Coyote Cafe and Geronimo have elk tenderloin, but that's hardly exotic by the standards you're outlining. They're also fairly stratospherically priced.
Jambo Cafe, an interesting African and Caribbean restaurant outside the tourist center, has some goat stews and the like on the menu.
Mariscos la Playa occasionally does semi-exotic seafood like shark fajitas.
Shohko Cafe routinely gets really interesting exotic fish in for their sushi and sashimi. House-smoked pike, kohada, needlefish, etc. It's also a remarkably good Japanese restaurant even by coastal standards.
Horsemen's Haven dishes out traditional NM-style food, but routinely makes it about as spicy as I've had it around town. Their green chile is spicy enough to border on punishing.
La Casa Vieja about an hour south in Corrales does a bacon martini and a jalapeno-infused margarita (the latter of which I've had and enjoyed), but they're currently closed due to structural issues with the building.
Also south in Albuquerque is a very good, quite authentic Chinese/Taiwanese restaurant named Budai. They have a "secret" menu available on request that has all the fun stuff on it, tripe, pig's ear, etc.
Of the above restaurants, the ones I think have the best chow are Jambo, Shohko, Mariscos la Playa, and Budai.
321 Johnson St, Santa Fe, NM 87501
4541 Corrales RD, Corrales, NM 87048
6300 San Mateo Blvd NE, Albuquerque, NM 87109