Preheat oven to 400°F. Line a baking sheet with parchment paper.
For puffs: In a saucepan, heat butter and 1 cup water to a boil. Add flour and stir vigorously over low heat, until mixture forms a ball. Remove from heat. Beat in eggs all at once and continue beating until smooth. Drop dough in balls onto prepared baking sheet. Bake for 35 to 40 minutes, or until puffy and golden brown. Cool.
For filling: In a bowl, combine sugar, flour, and salt.
In another bowl, beat eggs. Add to flour mixture and stir until well combined.
In a saucepan, heat milk to almost boiling (until milk forms bubbles at edge of pan.) Add milk, a small ladleful at a time, to flour mixture and whisk vigorously. Repeat, using about half of milk. Combine flour mixture and remaining milk in saucepan and whisk to blend. Bring to a boil, stirring constantly. When mixture is thick, like pudding, it should also be just starting to boil. Remove from heat. Add vanilla and whisk to blend. Let cool.
Slice cooled puffs (horizontally or vertically) and add filling.
For topping: In a saucepan, melt chocolate and butter over low heat. Remove from heat and stir in hot water. Mix until smooth. Add confectioners’ sugar and mix until smooth. Drizzle over filled éclairs. Refrigerate.