Special equipment: a pastry bag and a large round tip
For the cream puff dough: Preheat the oven to 375 degrees F. Line two baking sheets with parchment or silicone baking mats.
Bring the butter, milk, sugar, salt and 1/2 cup water to a rapid simmer in a medium pan. Dump the flour in all at once. Stir with a wooden spoon over low heat. The dough will come together as a smooth ball and the bottom will have a skim of dough stuck to it.
Remove the dough from the pot and place in the bowl of stand mixer fitted with a paddle attachment. (I love these beater-blade paddles with the rubber sides, so you don't have to scrape down the bowl. If you don't have one, then scrape down the bowl after each egg.) Add the eggs one at a time, mixing well after each, until it comes together in a smooth paste. It will be thick enough to hold its shape, but thin enough to pipe easily.
Fit a pastry bag with a large round tip and pipe the dough into 2 1/2-inch logs on the baking sheets. If there are points sticking up on the ends just wet your finger and smooth them out. Bake until they are puffy but have little color, about 10 minutes. Quickly rotate the sheets from top to bottom and back to front. They may deflate a touch, but don't worry; they will puff again as they continue to bake. Bake just until they start to turn golden, an additional 7 to 10 minutes.
Prop the door open with a wooden spoon and continue to bake for about 5 more minutes. Remove from the oven and allow to cool completely before filling. They can be made ahead and frozen for up to a couple of weeks.
For the white chocolate pastry cream: In a pan, heat the milk and 1/4 cup of the sugar over medium heat until simmering. In a bowl, combine the remaining 1/4 cup sugar, the cornstarch, salt and egg yolks. Once the milk has come to a simmer, ladle a small amount out and whisk it into the egg mixture to warm it up. This is called tempering and it prevents the eggs from cooking too quickly.
Once the egg mixture is warm to the touch, whisk it into the pan of milk. Cook over medium heat until it comes to a boil; whisk it vigorously for about 3 minutes so that you are sure to cook the cornstarch. Whisk in the butter, rum if using and vanilla until it is smooth. Place in a large bowl and immediately whisk in the chopped white chocolate. Press plastic wrap directly onto the surface of the pastry cream to prevent a skin from forming. Refrigerate until well chilled.
Once you are ready to assemble the eclairs, cut the pastry open along the sides with a sharp knife. Gently stir the pastry cream and pipe it into the pastry shells. Refrigerate for at least 1 hour.
For the ganache: Heat the cream to a simmer. Add the chocolate, swirl the pan to cover the chocolate with the cream and allow to sit for 2 minutes. Gently stir the cream and chocolate together. You don't want to whip too much air into it or it won't be smooth and shiny.
Dip the chilled eclairs into the ganache.
Refrigerate until ready to serve (up to 8 hours) and then watch them disappear like....lightning!