Begin by blending pistachio paste with melted white chocolate until smooth, creating a luscious filling. Spoon the mixture into an ice cube tray or silicone molds and freeze for at least one hour.

While your filling sets, create a double boiler by placing a heatproof bowl over a pot of simmering water. Add the dark chocolate and unsalted butter to the bowl, stirring gently until melted and combined, about 5-7 minutes.

In a large mixing bowl, whisk together the eggs and sugar until thick and pale, about 3-4 minutes. Slowly fold in the melted chocolate mixture, followed by a pinch of salt and vanilla extract.

Preheat your oven to 425°F (220°C) and grease four ramekins with butter and a dusting of flour. Spoon half of the batter into each ramekin, then carefully place a frozen pistachio filling in the center, covering it with the remaining batter.

Place your ramekins on a baking tray and bake in the preheated oven for 12-14 minutes. Look for the edges to be set while the center remains slightly jiggly.

Remove the cakes from the oven and let them sit for 1-2 minutes before inverting onto plates. Serve the cakes warm to enjoy the gooey center.