Preheat oven to 450 degrees F. Line a large baking sheet or two medium baking sheets with parchment paper.
Partially melt the butter in a large 3-4 quart saucepan, then add the water and salt. Bring to a boil. Remove from heat briefly and stir in the flour all at once with a wooden spoon. Reduce heat to low and continue stirring until mixture leaves side of pan and forms a ball. Stir for another minute or two until mixture starts to leave a film on the bottom of the saucepan. This step is to remove any floury flavor.
Remove from stove and transfer dough to a mixing bowl. Mix gently with a mixer for 1 minute, then allow the mixture to cool so that it won't cook the eggs.
When the mixture is not so hot that it will cook the eggs, go ahead and crack in one egg. Mix with the mixer until egg is incorporated, then add remaining eggs one at a time beating to make a soft dough. You can also do this step with a whisk if you prefer.
Transfer mixture to a large piping bag fitted with a large round tip or use a spoon to make about 10 (you may get more) lines of dough each about 4 inches in length and somewhere a little over and inch wide.. You could also do this with a heavy duty zipper bag. Just put the dough in the zipper bag and snip off the bottom corner.
Bake at 450 for 15 minutes, then reduce heat to 325 and continue baking for another 15 20 minutes or until golden brown. As soon as you remove them from the oven, prick lightly with a fork to help steam escape. This is supposed to help keep the insides from being soggy. I think it helps a little, but I don't always remember to do it. Let cool while you make the filling.
Filling: Whisk together pudding mix and milk for about 2 minutes. When pudding thickens, fold in the whipped topping (or sweetened homemade whipped cream).
To fill the eclairs, use a Bismark tip (the long skinny one) and poke it into the pastry. Squeeze it in there and fill. You can also split the eclairs and fill, or hollow out with a poke and use a heavy duty freezer bag with the corner snipped off.
Keep the filled eclairs cold while you make the icing.
To make the icing, combine the chopped chocolate and butter in a saucepan and heat over medium low, stirring constantly, until melted and smooth. Add the sifted sugar gradually, stirring well (you'll have a thick paste), then add the vanilla and the 3 tablespoons of very hot water, stirring until smooth. Mixture will be runny while warm. Allow it to cool and thicken if needed.
Spoon over the cold eclairs and let set.