Pour 1 cup of water into the bottom of your Instant Pot to generate steam for cooking the cake.
In a microwave-safe bowl, combine the cubed unsalted butter and chopped semi-sweet chocolate. Microwave in 20-second intervals, stirring in between, until melted and smooth (about 1 minute). Allow to cool slightly.
Whisk together the powdered sugar and the melted chocolate mixture until well combined. Add the large egg, egg yolk, pure vanilla extract, and espresso powder. Mix until smooth.
Gently fold in the all-purpose flour and salt until just combined. Be careful not to overmix.
Grease silicone ramekins with butter or cooking spray. Pour the batter evenly into each ramekin, filling about 3/4 full.
Place the filled ramekins on the trivet inside the Instant Pot. Ensure they are stable and not touching the water.
Secure the Instant Pot lid with the steam release valve set to sealing. Select "Pressure Cook" or "Manual" and set the cooking time to 9 minutes.
Allow a natural pressure release for 5 minutes, then carefully switch the valve to venting to release any remaining steam.
Remove the ramekins carefully. Let cool for a minute, run a knife around the edges, invert onto a plate, and serve warm with ice cream or powdered sugar.