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.