Grease a 6-inch or 7-inch cake pan and line the bottom with parchment paper.
In a microwave-safe bowl or double boiler, melt the dark chocolate and butter together until smooth. Let cool slightly.
In a large bowl, whisk the eggs and granulated sugar until the mixture becomes lighter in color and slightly thickened.
Add the melted chocolate mixture to the eggs and whisk until combined.
Stir in the cocoa powder, vanilla extract, and salt until the batter is smooth.
Pour the batter into the prepared pan and smooth the top.
Preheat the air fryer to 320 degrees F if your model requires preheating.
Place the pan into the air fryer basket and air fry for 20 to 25 minutes, or until the edges are set and the center has a slight jiggle.
Remove the cake from the air fryer and let it cool in the pan for 15 minutes.
Transfer to a wire rack and cool completely before serving.