Preheat the oven to 425°F (220°C).

In a 4-quart saucepan, bring the butter, salt, and 1½ cups water to a boil over high heat.

Remove the pan from heat, add the flour all at once, and stir vigorously with a wooden spoon until the mixture forms a thick dough and pulls away from the sides of the pan, about 2 minutes.

Return the pan to heat and cook, stirring constantly, until the dough is lightly dried, about 2 minutes more.

Transfer the 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.

The dough should 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 the other spoon, scrape the dough onto a parchment-lined baking sheet, setting pieces 1 inch apart.

Lightly beat the remaining egg with a 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°F (175°C) and continue to bake until well browned, about 15 minutes.

Let cool.

In a 4-quart saucepan, bring 1 cup of milk and sugar to a boil over medium heat.

Meanwhile, whisk the remaining ½ cup milk, cornstarch, and egg yolks together in a large bowl.

Slowly pour half the hot milk into the yolk mixture, whisking constantly, then return the mixture to the saucepan.

Cook, stirring constantly with a wooden spoon, until it thickens and just returns to a boil.

Stir in the vanilla and transfer to a bowl; cover with plastic wrap and refrigerate until chilled.

In a large bowl, beat the softened butter on medium speed with a hand mixer until pale and fluffy.

Add the cold filling and beat until smooth and fluffy, about 4 minutes.

Spoon the filling into a pastry bag fitted with a plain ¼″ tip.

Gently poke a hole in the flat side of each baked, cooled puff with the tip and pipe in the filling.

Place 2 cups of sugar and ½ cup of water in a shallow saucepan and stir to combine.

Cover and cook over medium heat until the sugar turns light amber, about 15–20 minutes.

Remove from heat.

Using tongs, dip the top of each filled puff in hot caramel.

Place puffs, glazed side up, on a plastic-lined tray.

Form the base with 12–14 glazed cooled puffs, sticking them together with more caramel.

Add puffs, layer by layer, to form a hollow cone.

Reheat the caramel as needed if it becomes too thick, and make more caramel with the remaining sugar and water when the first batch becomes too thick to work with.

Allow the caramel to cool until it is the consistency of honey.

With a spoon, drizzle thin strings of caramel around the cone; let cool until brittle and set.

Serve the croquembouche within 4 hours of making to ensure the filling doesn't soften the puffs.