Prepare pastry cream: In a small bowl, dissolve cornstarch in 1/4C whole milk. Beat 1 whole egg into mixture, followed by 1 yolk at a time.
In a small saucepan over medium heat, bring remaining 3/4C whole milk and sugar to a boil, whisking. When mixture boils, slowly drizzle 1/3 milk mixture to egg mixture, whisking constantly. Return pan to medium-low heat, slowly adding egg mixture while whisking until mixture boils. Remove from heat, stir in butter and vanilla.
Strain mixture through a fine mesh strainer into a small bowl. Cover with plastic wrap, pressing to the surface of the pastry cream. Allow to cool to room temperature before chilling until ready to use. May be chilled for up to several days.
Prepare eclair shells: Preheat oven to 425°F. Line two baking sheets with parchment paper. Prepare a pastry bag with a large round tip.
In a medium saucepan over medium heat, stir together water, butter, sugar and milk. Stir occasionally until mixture boils. Remove from heat. Sift in flour. Return to heat, mixing in flour and stirring until dough pulls away from the sides of the saucepan.
Transfer eggs to the bowl of a stand mixer with paddle attachment. Mix on low to cool dough slightly. Begin adding 1 egg at a time, followed by cocoa powder. Mix until eggs and cocoa powder are fully incorporated into dough.
Spoon dough into prepared pastry bag. Pipe 1"x4" strips. Remove peaks by dabbing your finger in water and gently brushing the peaks.
In a small bowl, beat 1 whole egg with a pinch of salt. Brush egg wash over piped dough.
Bake for 10 minutes at 425°F, then reduce heat to 350°F and continue baking for 10 additional minutes.
Allow eclair shells to cool completely before filling with pastry cream.
Add pastry cream to a pastry bag fit with a long tip. Pierce one end of an eclair and fill with pastry cream. Watch eclair slightly puff up and slowly remove tip.
Coat top of eclairs with a drizzle of white chocolate, or you may dip the top side instead of drizzling. Best served immediately or store in an airtight container, chilled.