Mix flour and salt together in a large bowl; push into a mound and make a well in the center. Place egg, water, and olive oil into the well; mix together, slowly incorporating the flour mixture until dough is combined.
Turn out dough onto a floured work surface; knead until ball forms, 5 to 10 minutes. Divide in half; form into balls. Wrap each dough ball in plastic wrap; rest dough until malleable, about 30 minutes.
Remove plastic wrap. Roll out dough balls; cut into desired shapes with a knife or pasta roller.