I am very familar w/things Japanese and I have never heard of moochi cheese. I have however, am very familiar w/mochi.
Mochi is pounded rice. It's pounded until it looks like a white glutenous blob then hardened and formed into a disk shape. Mochi is then used to make Okonomiyaki, mochi balls for teppan yaki (Scewered) and other meals that call for such an item. It's very possible the Sake House is using Mochi to make a dish called "Mochi Cheese". Most any Japanese market will have mochi. I'm not on the westside but, when I google, these places "pop up" :
Mitsuwa Market Place Santa Monica 3760 Centinela Av
Nijiya Market 2130 S Sawtelle Bl #105 West Los Angeles
Any 99 ranch market should also carry mochi. If it's not in disk form, it may be in sticks.
Mochi is also seen as a "puff" style treat found at Whole Foods. This item is in plain, raisin, garlic and ??... Large squares for the toaster oven and found in the fridge section at the market. BTW- Japanese people on average don't eat "real" cheese. When I was in Japan, it was very popular to have fruit salad sandwiches w/cream cheese slices but,that's it.
Sake House serve sushi and robatayaki ("stuff grilled on skewers", more or less) -- no cheese. They do serve grilled mochi, though, and Kitchen Queen has done a good job explaining mochi.
I buy in disks at Nijiya and heat it up in my toaster oven until it's the right consistency.
re: Das Ubergeek
Sake House has a mochi cheese appetizer - sorry I typed moochi before when I was in a hurry. It's the mochi as described above with cheese inside and either fried or deep fried. A japanese friend told me its a very common pub food in Japan. It's very good you should check it out. I believe they buy it frozen because of a conversation I had with the waitress there.
I've never heard of mochi cheese, either. I'll bet what you ate was hanpen cheese.
Hanpen is a mildly flavored fish cake with a tender, spongy consistency. They come in square pieces of 3" x 3" x 1/2" thickness. I can understand the visual confusion with mochi, but unless what you ate had a stretchy, pull-y, sticky rice consistency, you weren't eating mochi.