Preheat your oven to 450° and brush the inside of both ramekins with melted butter. Sprinkle cocoa powder in each one and shake to coat all the sides. Tap out and discard any excess cocoa powder.
Into a mixing bowl, sift the powdered sugar and flour, set aside.
Set a large bowl over a pot of simmering water (it should be shallow water and NOT touch the underside of the bowl) add in the chocolate, butter and espresso powder. Stir until just melted. Use oven mitts to remove the bowl off of the pot and finish stirring until smooth.
To the chocolate, add the vanilla and kosher salt.
Then, in a separate bowl, whisk the eggs and egg yolks until thoroughly beaten. Should be light yellow and frothy.
To the bowl with the flour and sugar, pour in the melted chocolate and beated eggs. Stir until incorporated.
Divide the batter among the two prepared ramekins (a heaping 3/4 cup per ramekin) and place onto a small sheet pan.
Tap gently on the counter before sliding the pan onto the middle rack of your preheated oven. Bake on the middle rack for 12 minutes (for a more runny center) or 14 minutes (for a less runny middle).
Let cool for 5 minutes before running a knife around the inside of the ramekin. Flip a dessert plate and set it ontop of the ramekin. Use a kitchen towel so you don't burn your hands. Then quickly flip so the plate is right-side up and the cake slides out.
Dust with powdered sugar and serve with fresh berries.