Preheat the oven to 425 F and thoroughly grease 4 ramekins with butter, then dust the insides lightly with cocoa powder. Set aside.
In a large, microwave-proof bowl, combine the butter, dark chocolate, and white miso paste. Microwave the ingredients in 20 second increments, stirring regularly, until the ingredients have fully melted and combined. Set aside.
Separate 2 egg yolks from the egg whites, then add the 2 egg yolks to a bowl along with the remaining 2 whole eggs. Add the brown sugar, white sugar, and vanilla extract to the bowl and then whisk everything together for a few minutes until the egg mixture is pale and frothy.
Pour the chocolate mixture into the egg mixture, and then whisk to thoroughly combine.
Add the cocoa powder and the all-purpose flour to the bowl, and whisk these in until they are just combined to complete your batter.
Divide the batter between the 4 prepared ramekins, then place these into the oven for 10-11 minutes, until the outsides are set and the inside is still very wobbly.
Allow the cakes to cool for a few minutes, before gently inverting the ramekins and turning out the lava cakes onto serving plates. Finish off with a dusting of powdered sugar and fresh strawberries if desired, and then enjoy right away.