First, get your oven ready. Turn it on to 425°F (218°C) and let it heat up while you make the batter. Take out 4 ramekins (those small, round baking dishes) and brush them really well with melted butter. This is important so your cakes won't stick. Sprinkle a little flour inside each one and tap out the extra.
Get all your ingredients out and ready. You'll need butter, white chocolate, eggs, sugar, matcha powder, flour, vanilla, and salt. Make sure your eggs are at room temperature - this helps them mix better.
Put your butter and white chocolate in a bowl that can handle heat. Melt them together by placing the bowl over a pot with a little simmering water (don't let the bottom of the bowl touch the water). Stir until everything is smooth and melted. Take it off the heat and let it cool a bit while you work on the next step.
In another bowl, crack your 2 whole eggs and add 2 egg yolks. Pour in the sugar. Beat this mixture until it becomes pale and fluffy - about 3 to 4 minutes. You can use a hand mixer or whisk really well by hand.
Now comes the matcha part. Sift the matcha powder and flour together through a fine strainer to get rid of any lumps. Add this green mixture to your egg mixture along with the vanilla and salt. Stir just until they're mixed together.
Take your melted white chocolate and butter mixture (it should be warm but not hot) and slowly pour it into your green batter. Gently stir everything together until it's smooth and even in color.
Divide your batter between the buttered ramekins. Fill each one about ¾ full. At this point, you can bake them right away, or you can cover them with plastic wrap and put them in the fridge for up to 24 hours if you're planning ahead.
When you're ready to bake, put the ramekins on a baking sheet and slide them into your hot oven. Bake for 9 to 10 minutes. The edges should look set, but the middle should still be a bit jiggly. Don't overbake or you won't get the gooey center!
Let them cool for just 1 minute - no longer, or the centers will keep cooking. Run a knife around the edge of each cake, then put a plate on top of each ramekin and carefully flip it over. Lift off the ramekin, and your cake should come out easily.
Finish with a light dusting of powdered sugar on top if you like. Serve right away while they're warm and the centers are still flowing. You can add ice cream, whipped cream, or fresh berries on the side if you want to make it extra special.