For the pâte à choux: Heat oven to 425°. Bring butter, salt, and 1 1⁄2 cups water to a boil in a 4-qt. saucepan over high heat. Remove pan from heat, add flour all at once, and stir vigorously with a wooden spoon until mixture forms a thick dough and pulls away from sides of pan, about 2 minutes. Return pan to heat and cook, stirring constantly, until dough is lightly dried, about 2 minutes more. Transfer dough to a bowl, and let cool for 5 minutes; using a wooden spoon, beat in 8 eggs, one at a time, making sure each egg is completely incorporated before adding the next. Dough will come together and be thick, shiny, and smooth.
Dip two spoons in water, shake off excess, and scoop a walnut-size piece of dough with one spoon. With other spoon, scrape dough onto parchment-lined baking sheet, setting pieces 1″ apart on a baking sheet. Lightly beat remaining egg with pinch of salt and brush each piece of dough with it. Bake until puffed and light brown, about 10 minutes. Reduce oven temperature to 350°, and continue to bake until well browned, about 15 minutes. Let cool.
Spoon pastry cream into a pastry bag fitted with a plain 1⁄4″ tip. Gently poke a hole in the flat side of each baked, cooled puff with tip and pipe in filling.
For the caramel: Place 2 cups sugar and 1⁄2 cup water in a shallow saucepan and stir to combine. Cover and cook over medium heat until sugar turns light amber, about 15–20 minutes. Remove from heat.
Using tongs, dip top of filled puffs in hot caramel. Place puffs, glazed side up, on a plastic-lined tray. Form base with 12–14 glazed, cooled puffs, sticking them together with more caramel. Add puffs, layer by layer, to form a hollow cone. (Reheat caramel until liquid again if it becomes too thick; repeat making more caramel with remaining sugar and 1⁄2 cup water when first batch of caramel becomes too thick to work with.) Allow caramel to cool until it is the consistency of honey. With a spoon, drizzle thin strings of caramel around cone; let cool until brittle and set. Serve croquembouche within 4 hours of making to ensure the filling doesn't soften the puffs.