Whisk eggs together with sugar and salt. Add cheese, vanilla extract and baking powder to the mixture, and mix until you get an even mixture.
Using spatula, in thirds, fold in flour into the mixture.
Preheat oil in deep frying pan to medium low, be sure not to heat oil on high, otherwise doughnuts will cook on the outside and be raw on the inside.
With a spoon or an ice cream scoop, scoop out a small round portion of the dough, drop it in preheated oil. Try leaving space between each one in a frying pan. Flip them with a fork until each side is golden brown.
Carefully lift each doughnut out onto racks lined with paper towels. Optional: sprinkle with powdered sugar.