Set a pie plate in the freezer to chill. Generously butter four 6-ounce ramekins. Sprinkle cocoa powder over the butter, tap out excess, and set aside.
Combine chocolate and butter in a large heatproof bowl. Set the bowl over a saucepan of barely simmering water, stirring frequently until melted and smooth. Remove from heat.
Combine raspberry puree, 2 teaspoons of the sugar, and the vanilla extract in a small bowl. Transfer 4 tablespoons of the melted chocolate to the raspberry mixture and stir until smooth.
Spread the chocolate raspberry mixture into the chilled pie plate and set it back in the freezer to firm, about 10 minutes.
When the chocolate-raspberry mixture is cold enough to hold a shape, use a melon baller or teaspoon to scrape it into 4 small balls (it’s okay if they’re uneven and rough looking). Leave the balls in the pie dish and return them to the freezer.
Whisk the eggs, remaining sugar, and salt in the bowl of a standing electric mixer on medium-high speed until they triple in volume and turn very pale yellow, about 8 minutes.
Fold about one-quarter of the egg mixture into the melted chocolate along with the cocoa powder. Add the eggs to the chocolate mixture in two more additions, folding until evenly mixed.
Using a large ice cream scoop, add a generous dollop of batter to each of the prepared ramekins. Press a chocolate-raspberry mound gently into the batter in the center of each cup. Add the remaining batter, making sure the ganache centers are completely covered. Tap the ramekins on the counter several times or smooth the tops with the back of a spoon to level the batter. Cover the cups with plastic wrap and refrigerate for at least 3 hours.
Twenty minutes before you plan to serve the cakes, remove the cups from the refrigerator and let sit, uncovered, at room temperature. In the meantime, preheat oven to 400° F.
Set the cups on a baking sheet and bake for 12 to 14 minutes, until the cakes are puffed and the centers are just set. Let cool for about 3 minutes.
Run a thin spatula around the sides of each cup, taking care as the ramekins will be quite hot. Working one at a time, set a ramekin on top of a clean kitchen towel. Set a plate top-side down over the ramekin. Gather the towel around the ramekin as you hold it tightly against the plate. Flip both over and pull the ramekin straight up to release the cake onto the plate.
Serve the cakes hot with scoops of vanilla ice cream and fresh raspberries.