Papanasi (pronounced "papanash") are traditional Romanian cheese sweets. I grew up in a Romanian Jewish family. Most of my childhood's cuisine was based on Romanian dishes, mostly meat, potatoes and eggplants but there were also cheese dishes that I loved – savory and sweet.
Since I became vegetarian, 22 years ago, I've been focusing on those vegetable and dairy dishes. One of the dairy sweets that both my grandmother and mother used to make (and still does, G'd bless her) is called Papanash. The original Papanash that you can find in most Romanian Restaurants is a sweet cheese DOUGHNUT that looks a little similar to the American doughnuts we're accustomed to (as oppose to Hanukka's doughnuts that do not have a hole and are filled with jam or other fillings), except the Papanash doughnuts do not come out as round as American doughnuts because their dough is softer.
In my house we used to eat two other kinds of Papanash, though: cheese patties and cooked cheese and semolina dumplings. The original doughnut version was introduced to me much later, when I started inquiring about Romanian cuisine.
The original Papanash: Cheese Doughnuts
Romanians do not waste anything so they deep fry the round dough that was cut out of the doughnut's center and place it on top of the Papansh that is served with sour cherries jam and sour cream or whipped cream.
Here are the two other versions I'm used to:
Papanash are usually sweet. However, I took my sweet recipe and turned it into a savory dish. You can serve them with the tomato sauce or without it. They're great anyway.
In Israel we have soft but very dry low-fat cheeses for these kinds of dishes. I'm not familiar with American cheeses but the best substitute I can think of is Ricotta cheese. Make sure you use a low-fat soft but DRY cheese.