Combine yeast and sugar in a large bowl. Then add the eggs and water.
Once the mixture is bubbling, add the salt and flour. The dough will be very wet. After you add your flour, stir it well, but don’t worry if there are some lumps. Finally, beat in the softened butter until the dough is smooth and very soft. If it’s really sticky, add more flour by the handful.
Let this dough rise in the bowl at room temperature for at least 2 hours. It will at least double in volume.
You can punch it down a bit at this point and can use it right away or store it in the fridge for up to 7 days (in airtight container).
When you ready to make the beignets, dust the dough with some flour.
Roll the dough out until it’s 1/8-inch thick. Use more flour if the dough is sticking. Then take a knife or, even easier, a pizza cutter, and slice up the dough into 2X2 inch squares. Let the dough rest for 10 minutes after you cut it.
Preheat oil to 350 degrees in a large pot or deep-fryer. Working with a few beignets at a time, add a small amount of peach filling to each square and fold it over itself, pinching the edges. Fry dough squares in batches for about 2-3 minutes per side until they are golden brown.
Take them out with a slotted spoon and transfer to a paper towel to drain quickly. Then dust with powdered sugar. Use a heavy hand here.
For peach filling:
Dunk peaches in boiling water or 45 seconds. Remove and rinse with cold water. Carefully peel peaches and dice. Toss diced peaches in a colander with sugar and a pinch of salt and sit aside for later.