How to Make It
Preheat the oven to 400°. In a medium saucepan, combine the milk with the butter, vanilla and salt and bring to a boil over moderately high heat. Add the flour and, using a wooden spoon, beat until a smooth dough forms. Reduce the heat to low and cook the dough for 3 minutes, stirring constantly to dry it out. Remove the pan from the heat and beat in the 5 eggs, 1 at a time.
Line 2 rimmed baking sheets with parchment paper. Drop 1 1/2-tablespoon-size mounds of batter onto the baking sheets, spacing them 2 inches apart. You will need 24 puffs. Brush the puffs with the beaten egg wash and bake on the 2 lower oven racks for 30 minutes. Reduce the oven temperature to 250° and bake for about 20 minutes longer, or until the puffs are browned and dry. Let the puffs cool to room temperature on the baking sheets.
Raise the oven temperature to 325°. Lightly oil a large rimmed baking sheet. In a medium bowl, stir the white sugar into the water until dissolved. Stir in the pistachios, then the turbinado sugar. Spread the coated nuts on the baking sheet in an even layer and bake for about 10 minutes, or until shiny. Let the candied nuts cool completely on the baking sheet, then break apart.
In a medium saucepan, combine the sugar, cinnamon stick and corn syrup. Bring to a boil over moderate heat, stirring a few times to dissolve the sugar, and simmer until a deeply colored caramel forms, about 10 minutes. Meanwhile, in a saucepan, bring the cream just to a simmer. Remove the caramel from the heat. Slowly and carefully pour in the hot cream, stirring constantly; the caramel will bubble up, but keep stirring until the bubbles subside. Return the pan to the heat and stir to dissolve any hardened bits of caramel. Remove the pan from the heat, stir in the chocolate and let stand for a few minutes to melt. Discard the cinnamon stick and whisk the sauce until smooth.
Cut each puff in half and set 3 bottom halves on each plate. Generously fill the profiteroles with ice cream and cover with the puff tops. Pour the caramel-chocolate sauce over the puffs, sprinkle with the candied pistachios and serve immediately.