For the Praline Paste: Lightly grease a rimmed baking sheet (quarter-size is ideal, but half-size will also work) and set aside. In a 2-quart stainless steel saucepan or saucier, combine water, sugar, corn syrup, hazelnuts, almonds, and salt. Cook over medium heat, stirring frequently with a heat-resistant spatula to ensure sugar dissolves and doesn't scorch, until mixture darkens to a tawny brown, and registers 320°F (160°C) on a clip-on digital thermometer, about 10 minutes.

Remove from heat, then immediately stir in baking soda, folding with heat-resistant spatula until candy is evenly foamy, then pour onto prepared baking sheet. Let cool to room temperature, about 30 minutes, then proceed immediately to the next step (if praline is allowed to sit out for an extended period of time after cooling, it will turn tacky and soft).

Break praline into pieces and transfer to bowl of a food processor. Pulse several times to pulverize candy, then let food processor run continuously until praline transforms into a thick, peanut butter–like paste, about 3 minutes. Transfer paste to a small bowl, cover, and set aside.

For the Choux: Adjust oven rack to middle position and preheat to 375°F (190°C). Using a pen or pencil, trace an 8-inch circle in the center of a 12 by 18-inch sheet of parchment paper. Flip parchment over so that drawn ring is on underside and set in a rimmed baking sheet; set aside. Pipe a small amount of choux paste under each corner of parchment paper lining the baking sheet (the dough acts as a glue and keeps the paper in place as you pipe).

Holding filled pastry bag at a 90° angle, apply steady downward pressure and slowly pipe first ring of choux along traced circle. To stop piping, cease applying pressure and swirl pastry tip away from piped ring. Pipe a second ring of choux just inside first ring, making sure that the two rings are touching one another. Pipe a third, final ring of choux on top of, and nestling in the groove between, the two piped rings. To smooth out surface of dough, dip a finger into cold water and gently pat down any bumps.

Using a pastry brush, gently brush a light layer of egg wash over surface of choux dough, being careful not to let excess egg wash drip down onto the parchment. Sprinkle sliced almonds in an even layer over top.

Bake until choux ring is puffed, deeply golden brown, and feels hollow when carefully lifted from parchment with a spatula, about 35 minutes. Turn off oven, partially open oven door, and let stand for 30 minutes to allow choux ring to dry and fully set. Remove from oven and let cool completely, about 15 minutes.

For the Crème Mousseline: Meanwhile, in bowl of a stand mixer fitted with a paddle attachment, beat softened butter on medium speed until fluffy and light, about 5 minutes. Scrape down bowl and beater with a flexible spatula. Add prepared praline paste and beat on medium speed until well combined, about 3 minutes.

With mixer running at medium speed, add pastry cream, a few tablespoons at a time, pausing to scrape down bowl and paddle as needed, until thoroughly incorporated, about 30 seconds.

Switch to whisk attachment and whip mixture on medium speed until crème mousseline is soft and airy, about 3 minutes (see notes). Transfer crème mousseline to a pastry bag fitted with a ¾-inch star tip.

To Assemble the Paris–Brest: Transfer cooled choux ring to a large cutting board. Holding a serrated knife parallel to board, make a small cut from the outside and slide it all the way through so tip comes out inside the ring. Gently slice ring in half horizontally, using a sawing motion while simultaneously rotating choux ring, to yield a top and bottom half. Carefully lift off top half and place it next to bottom half.

Pipe crème mousseline in an even 2-inch-high layer onto cut side of bottom half of choux ring, making sure to apply steady pressure while piping. Place top half of choux ring over piped filling, and dust top evenly with powdered sugar. Slice into portions with serrated knife, and serve.