To make the nougat:
Line a 9 x 13 inch baking pan with parchment paper, leaving some overhang on both sides. Spray the paper with oil, making sure to cover all areas of the pan. Set aside.
In a medium bowl, whisk together the soy milk powder and powdered sugar. Sift the mixture through a fine-mesh strainer to remove any lumps. Set aside.
In a small microwave-safe bowl, combine the dark chocolate and cocoa butter. Microwave in 30-second intervals, stirring in between, until melted and smooth. Set aside to cool slightly.
In the bowl of a stand mixer fitted with a whisk attachment, or in a large bowl using a hand mixer, combine the aquafaba, cream of tartar, and 1/4 cup (60 g) of the corn syrup. Set aside.
In a large heavy-bottomed saucepan over medium-high heat, combine the remaining 2 3/4 cups (650 g) of corn syrup, granulated sugar, and water. Stir until the sugar dissolves and the mixture comes to a boil.
Stop stirring and attach a candy thermometer to the side of the pan. Brush down the sides of the pan with a wet pastry brush to remove any sugar crystals. Cook until the temperature reaches 252°F (122°C), which is hard-ball stage.
When the sugar mixture reaches about 230°F (110°C), start whipping the aquafaba mixture on high speed until it forms medium-firm peaks. If it reaches this stage before the sugar is ready, reduce the speed to low and keep it running.
When the sugar mixture reaches 252°F (122°C), carefully and slowly pour it into the aquafaba mixture in a thin stream, while continuing to whip on high speed. Avoid hitting the whisk or the sides of the bowl with the hot syrup.
Once all the syrup is added, keep whipping on high speed until the mixture is thick, fluffy, and less glossy, about 3 minutes. Let it cool slightly for a few minutes.
Using a rubber spatula, fold in the soy milk powder mixture until well incorporated. The mixture will be very sticky and stiff at this point.
Fold in the chocolate mixture until evenly distributed. The mixture will become even thicker and harder to stir.
Transfer the mixture to the prepared pan and spread it evenly with an offset spatula or your hands. You may need to wet your hands or spatula to prevent sticking.
Let the nougat set at room temperature until firm but slightly soft when pressed, about an hour.
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