For pastry cream, bring milk and scraped seeds and pod of vanilla bean to a simmer over medium heat. In a bowl, whisk together egg yolks, sugar, cornstarch and salt. Whisking egg mixture constantly, gradually add milk, a ladle at a time, to the egg mixture until all the milk has been added. Pour mixture back in pot and return to medium heat. Whisking vigorously, bring mixture up to a simmer. Due to the cornstarch, pastry cream will thicken quickly, about 4 minutes. Remove from heat and strain. Stir in butter (this will help halt the cooking process). Stir in brandy, cover surface of pastry cream with plastic wrap and chill completely. Whip cream to medium peak and fold in. Chill until ready to assemble.
To prepare éclairs, preheat oven to 400° F. Heat milk, water, butter, sugar and salt and bring up to a simmer. Sift flour and add all at once. Stir mixture over medium heat for 2 minutes. By hand or with a mixer fitted with the paddle attachment, add eggs one at a time, blending completely before adding additional egg. To know if you should add the sixth egg, lift a spoon (or the paddle) from the paste. If it "sheets" or leaves batter as you lift it then it does not need egg. If it still feels tight (is hard to pull away) then add the extra egg.
Fit a pastry bag with a large plain piping tip. Fill bag and pipe 4-inch long lines of paste onto a parchment-lined baking sheet, leaving at least 2 inches between each (secure the edges of the parchment to the pan by sticking it with a little of the paste). After all the dough has been piped, touch the finishing point of each éclair with a finger dipped in water, to flatten the point the tip has left.
Bake éclairs for 12 minutes at 400° F. Do not open oven yet! Reduce heat to 360° F and bake for an additional 8 minutes, until éclairs are rich golden brown. Allow to cool before filling.
To fill, fill piping bag with plain tip with chilled pastry cream. Insert tip into a long end of éclair and fill until cream stops at end of éclair. Continue filling all éclairs.
For chocolate glaze, stir together icing sugar and cocoa powder. Stir in corn syrup with milk until a smooth, pourable consistency. Dip éclair tops in glaze and let set on a cooling rack to dry. Keep éclairs chilled until ready to serve