What to do with mochi...this version is white, hard and puck-like in shape?

I've actually found this post about mochi. They show the cakes that I've seen in the past and also the shape that I mentioned in my post.

It seems like you can:

1. grill them - they will brown a bit and rise.
2. boil them - they won't rise
3. microwave them

And the ways to serve it is with:

1. Butter+soy sauce (an interesting combo that I will try)
2. Rolled in ground sesame seeds + sugar or salt.
3. Rolled kinako (toasted soybean powder + sugar.)

UPDATE

Boiled

I cut them into about 1 inch x 1 inch pieces and threw them in boiling water. Boiled them for about a minute or so...then removed them to a bowl of cold water to stop the cooking process.

Dressings

I tried butter+soy sauce (1 TBSP melted : 1/4 tsp soy sauce) - It's actually pretty good!

I also ground up some black sesame seeds with sugar (1 TBSP : 1/2 tsp sugar)...this was actually my favorite of the two.

There are different version of mochi that serves refrigerated, room temperature, steam, shabu-shabu (hot pot), grill etc. From the shape, size and hardness you describe, it seems to me the type you're having are good for grilling on charcoal with flavoring as simple as oil and soy sauce.

[EDIT]
Rice cake can be cooked and served differently across asian cuisine. There is Chinese version of "NianGao": can be plainly fried or steam (sweet or salty), stir cry (salty), or boil in soup or hotpot (salty). The Koreans have "Dok". A popular way of cooking is with red spicy gravy or "DokBokGi". Korean also serve "Dok" in salty soup such as "Dok Goog" or "Dok Man Du Goog" etc, and sometimes serve as sweeten desert or appetizer that simply named "Dok" that comes in varieties of color and nuts filling.
The Japanese have the "MoChi": as desert the Japanese transform the MoChi to anything you can possibly imagine (with nuts, cracker, meat etc.), in cooking they grill, boil, hotpot etc, and even Mochi ice-cream.

My aunt is married to a Japanese person, and she eats mochi with seaweed, soy sauce, cheese and pork/chicken floss.

Here are the steps:

1. Bake the mochi until it puffs up, then dip both sides with soy sauce.
2. Put it on top of a piece of seaweed (for shushi/ onigiri). Then add cheese (I use sliced cheese but any preferable cheese can be used).
3. Put the chicken floss on top of cheese (any floss will do).
4. Wrap it up

What you are describing sounds like the very basic, traditional New Year's mochi—what you get when you hand-pound mochi and then hand-shape it. I've participated in making these a couple of times, and we used to get a batch every year from my auntie, when her JACL post pounded them for the holidays. I've also bought them frozen and made my own with a stand-mixer.

The way I learned to prepare them from my dad is to skillet-fry them in a little oil, covered and over fairly low heat. When you get the temperature just right, fresh mochi prepared this way will brown and puff up like a balloon. When you take it off the heat, it will deflate to a thin, crispy shell with a gooey, sticky center. Mochi that has dried out some will puff up less, but should still crisp. We serve with shoyu (soy sauce) or cinnamon and sugar, but the point of mochi is to be an excellent platform for other flavors, so feel free to experiment.

More recently, I've discovered the moffle, and it has become one of my favorite ways to prepare mochi. Thaw the "puck" if it is frozen, pop it in a hot, greased waffle iron, cook for a few minutes (I wait until the steam coming out slows down), and voila! perfect crispy/chewy mochi-waffles. Serve it with your favorite toppings, or try it with a frozen confection for a different version of mochi ice cream. As with the fried version, really fresh mochi will work better than old, dried-out mochi, but this is a great way to disguise the imperfections of my mixer-made mochi.

_{I realize this question is a few years old, but I just ran across it and figured with the holidays coming more folks might be getting gifts of this kind of mochi and wondering what to do with it.}

I'm Chinese and my mom's made something similar as well. I can't recall the name, but we've had it two ways:

Boiled in a soup, usually with a saltier broth, as the large 'mochi' block tends to be flavorless, it serves to balance what might otherwise be an overly salty dish.

Pan fried or deep fried, this method will make it puff up more like your link says. I've usually had it this way sweet. We only sprinkled on white sugar after frying, but now I realize there are several things you could do to add more complexity...sprinkle with sugar and brulee with a torch for extra texture, top with fruit and condensed milk, or serve it alongside ice cream instead of the traditional western pastry/pie.

When I was a child, I would eat this mochi baked, puffed up and dipped in a soy/sugar mix. Now all I can find are the blocks - I put them in water, microwave about a minute, then bake, dip in a soy/artificial sweetener mix and it's just like when I was a child.

Yes, baking them is a right way for this kind of mochi.

A lovely version I know for making mochi: lightly oil an oven-safe glass pan so mochi does not stick too much. If mochi is frozen, defrost it in the fridge for a few hours. Break the smaller squares apart, there are usually 6 or 8 of them. Place them in the oiled pan and place in oven about 300/350 degrees F and bake for a couple of minutes (6-10). Watch them. When the squares puff up, they are ready. Eat them by filling them with jelly, nut butters, savory mashes or even salad.

I learned to eat it as a soup type dish. My grandma is Japanese and I learned from my dad... it’s a traditional dish and people usually eat it during New Years and for birthdays.

You use dashinomoto powder mix, soy sauce and some water, and that makes the base. Boil the mochi in water until they float. We then add boiled spinach, tofu and some komoboko (steamed fish cake) pieces and then prep a bowl of the base add all the ingredients and then add the boiled mochi and it’s amazing.