Place whole eggs in mixer bowl and run hot tap water over them to warm.
Bring water, salt and butter to boil in a small to medium heavy bottomed pot. Cut flame. Add flour all at once and stir vigorously until dough pulls away from the sides of the pan and forms a ball. (one minute)
Drain eggs and wipe bowl dry.
Transfer dough to the warm bowl fitted with the paddle attachment and completely mix in one egg at a time. Scrape bottom of the bowl and re-mix thoroughly. Add the cheese and mix until just combined.
Spoon the batter into a pastry bag fitted with a 1/2 inch opening, (round or star tip). Pipe 8 circles, 3 inches wide by 1 inch high, on a sheet pan lined with parchment. Leave sufficient room between them for the gougeres to double in size.
Bake at 375 degrees for about 25 minutes or until evenly golden.
