Preheat your oven to 425°F (220°C). This high temperature is key to achieving that deliciously gooey center.
Grease the inside of each ramekin with a little butter and dust with flour, tapping out the excess.
In a double boiler or a microwave-safe bowl, combine the unsalted butter and semi-sweet chocolate. If using a microwave, heat in 30-second intervals, stirring in between until smooth and melted. Stir in the espresso powder until fully incorporated.
In a separate bowl, whisk together the light brown sugar and confectioners sugar until well combined.
Add the large egg, egg yolk, and vanilla extract to the sugar mixture. Whisk until smooth and well blended.
Pour the melted chocolate mixture into the egg and sugar mixture. Stir gently until everything is combined.
Sift in the flour and add a tiny pinch of salt. Fold gently with a spatula until just combined.
Divide the batter evenly between the prepared ramekins, filling each about 3/4 full.
Place the ramekins on a baking sheet and bake in the preheated oven for about 12-14 minutes, or until the edges are set but the center remains soft.
Remove the ramekins from the oven and let them cool for a minute. Carefully run a knife around the edges to loosen the cakes, then invert onto plates. Serve immediately.
