Cut the 1/4 cup unsalted butter into cubes. Use a few cubes to butter the insides of your ramekins and set the buttered ramekins aside. If you want to dust the finished cakes, set aside a small amount of the 1/2 cup powdered sugar now; otherwise use all of it in the batter. Place the Instant Pot trivet inside the pot and add 1 cup water.
In a medium heatproof bowl, combine the remaining butter cubes and the 4 oz. chocolate. Microwave in 15-second intervals, stirring between each interval, until the butter and chocolate are fully melted and the mixture is smooth. (Alternatively, melt over a double boiler.) Let the melted mixture sit 1–2 minutes to cool slightly so it is warm, not piping hot.
Whisk the 1/2 cup powdered sugar (minus any small amount you set aside for dusting) and the 1/4 teaspoon espresso powder into the warm chocolate until thick and smooth.
Whisk in the 1 large egg, the 1 egg yolk, and the 1/2 teaspoon vanilla extract until the mixture is fully combined. If the chocolate mixture feels very hot, temper the eggs by whisking a spoonful of the chocolate into the eggs first, then whisking the egg mixture back into the bowl.
Add the 1/4 cup all-purpose flour and the 1/4 teaspoon salt. Fold them in with a spatula until just combined; do not overmix.
Divide the batter evenly among your prepared ramekins. Place the filled ramekins on the trivet in the Instant Pot (arrange so they sit steadily).
Close the Instant Pot lid and set the steam valve to the "Sealing" position. Cook on High Pressure for 8 minutes.
When the cook time ends, press the "Cancel/Off" button and perform a quick pressure release carefully (keep your face and hands away from the steam).
Open the lid, put on oven mitts, and carefully remove the ramekins from the Instant Pot.
Run a thin knife around the inside edge of each ramekin to loosen the cake. Place a plate over a ramekin, invert the ramekin onto the plate, tap gently, and lift the ramekin off to release the cake. Let the cakes rest on the plate a minute to settle.
If you set aside any powdered sugar, lightly dust the tops now. Serve the lava cakes warm.