Step 1
Combine milk butter and salt in a medium saucepan. Scrape vanilla seeds into the mixture. Bring to a simmer over medium heat about 10 minutes.
Step 2
Meanwhile whisk egg yolks and sugar in a separate bowl until pale and light about 3 minutes and whisk in cornstarch until combined.
Step 3
Slowly add the hot simmered milk to the egg yolks whisking constantly to temper the yolks. Pour tempered mixture back into the pan and cook whisking constantly until custard is thick and holds whisk marks about 2 minutes. Strain custard through a fine-mesh strainer discard vanilla bean pods and place plastic wrap directly on the surface of the custard. Refrigerate for at least 2 hours.
Step 4
Preheat the oven to 425 degrees F (220 degrees C). Line 5 or 6 rimmed baking sheets with parchment paper.
Step 5
Combine milk water 1 cup plus 2 tablespoons butter sugar and salt in a large saucepot over medium heat; bring to a boil then immediately remove from heat and add the flour all at once. Stir slowly at first to combine then vigorously until mixture forms a uniform paste.
Step 6
Transfer paste to the bowl of a stand mixer fitted with a paddle attachment. Add 12 eggs 1 at a time blending on low speed until each egg is completely incorporated. Mixture may appear broken but keep stirring until it is glossy smooth and stretchy.
Step 7
Transfer dough into a pastry bag fitted with a 1/2-inch tip and pipe 1 1/2-inch puffs about 1 inch apart on the prepared baking sheets to get about 20 per sheet. Use a wet finger to smooth the tops. Beat the final egg with a splash of water in a small bowl; brush the tops of the puffs with egg wash.
Step 8
Place 2 to 3 baking sheets in the preheated oven and bake puffs for 15 minutes. Do not open the oven door or the puffs will deflate. Decrease oven temperature to 350 degrees F (175 degrees C) and bake until deeply golden brown about 10 minutes more. Remove puffs from oven and place sheet on a wire rack to cool.
Step 9
Increase oven temperature to 425 degrees F (220 degrees C) to continue baking remaining puffs.
Step 9
Make caramel while puffs cool. Bring sugar water and corn syrup to a boil in a medium saucepan over medium-high heat. Swirl pan occasionally to help dissolve the sugar but do not stir until syrup turns deep amber about 20 minutes. Meanwhile prepare a large bowl of ice water.