In the bowl of an electric stand mixer, combine the water, yeast and sugar. Allow to sit until yeast dissolves and starts to bloom, about 5 minutes. With the mixer's paddle attachment, stir in the flour, salt, 1/3 cup oil and egg until well combined.
Turn dough onto a lightly floured surface and roll to 1/4-inch thickness. Using a doughnut cutter, cut the dough into rounds, reserving the holes for frying, also. Allow cut doughnuts to rise until doubled in bulk, about 1 hour.
Pour vegetable oil into a large cast iron skillet or Dutch oven to a depth of 2 inches and heat to 375 degrees. Fry doughnuts in batches until puffy and browned on both sides, about 1 minute per side. Use a slotted spoon to transfer to a cooling rack.
Sprinkle warm doughnuts with powdered sugar. Serve immediately.
Notes
These doughnuts would be delicious with any number of toppings or glazes: traditional powdered sugar glaze, chocolate ganache and some sprinkles would be perfect, too!