Melt the dark chocolate, milk chocolate and butter in a microwave in 20-second intervals, stirring after each, or in a small saucepan over low heat. Stir until smooth.
Whisk the eggs and sugar in a large bowl until smooth and well combined. Gradually whisk in the melted chocolate mixture.
Add the flour, cocoa powder, baking powder and salt. Stir just until no dry streaks remain. Refrigerate the dough for 30 minutes.
Heat the oven to 350°F/180°C. Line a baking sheet with parchment paper.
Scoop the dough into 12 to 14 portions and roll into balls. Arrange on the prepared baking sheet, leaving about 2 inches/5 centimeters between them.
Sprinkle lightly with flaky sea salt.
Bake until the cookies are cracked on top and set around the edges but still very soft in the centers, about 10 minutes.
While the cookies are hot, pipe chocolate-hazelnut spread into the centers. Serve warm or let cool until just set.