Add eggs, one at a time, beating well after each addition until incorporated. Using a spoon or pastry bag fitted with a No. 10 or larger tip, spoon or pipe dough onto the prepared cookie sheet in 1 1/2×4-inch strips.
Bake in the preheated oven for 15 minutes. Reduce heat to 325 degrees F (165 degrees C) and continue baking until the bottoms sound hollow when lightly tapped, about 20 more minutes. Cool completely on a wire rack.
Make filling: Combine milk and pudding mix in a medium bowl according to package directions. Beat heavy cream with an electric mixer in a separate medium bowl until soft peaks form. Beat in confectioners’ sugar and vanilla. Fold whipped cream into pudding.
Cut tops off of cooled pastry shells with a sharp knife. Fill shells with pudding mixture and replace tops.
Make icing: Melt chocolate and butter in a medium saucepan over low heat. Stir in confectioners’ sugar and vanilla. Stir in hot water, 1 tablespoon at a time, until icing is smooth and has reached desired consistency. Remove from heat and cool slightly.
Drizzle chocolate icing over filled eclairs. Store in the refrigerator until serving.
