How to Make It
Preheat oven to 425°F. Combine Brussels sprouts, oil, kosher salt and pepper on a rimmed baking sheet; toss to coat. Bake until tender and browned in spots, 15 to 20 minutes, tossing halfway through cook time. Cool completely, about 30 minutes.
Skewer each Brussels sprout, inserting a toothpick piece into the stem end, leaving about 3/4-inch of the toothpick sticking out. Arrange Brussels sprouts on the Styrofoam cone by pressing the toothpick end into the cone. Set cone on a parchment lined cake stand and cover table underneath with a large sheet of parchment paper.
Bring sugar, water and corn syrup to a simmer in a small saucepan over medium-high. Cook, swirling pan occasionally (do not stir) until caramel turns a light amber color and registers 375°F on an instant read thermometer, 10 to 12 minutes. Remove from heat and swirl pan constantly on a heat proof surface until caramel thickens slightly, darkens and cools to about 280°F, about 4 minutes. Dip the tines of 2 forks in the caramel and lift: A thin thread of caramel will form. (If caramel drips instead of pulling into a thread, continue cooling.) Working quickly, swirl forks in circles around croquembouche, allowing threads of caramel to stick to Brussels sprouts, re-dipping forks in caramel as needed. Repeat process until croquembouche is covered in spun sugar (you will have leftover caramel). Caramel stays at the optimum consistency for about 4 minutes. Sprinkle croquembouche with flaky salt and carefully slide parchment paper out from underneath. Croquembouche will keep at room temperature for up to 3 hours.