Preheat the oven to 325°F (160°C).
In a food processor, blitz the cookies until finely ground.
Melt the butter in an 11-inch (28 cm) ovenproof skillet over low heat. Turn the heat off, tip the cookie crumbs into the skillet, and mix well. Then spread and pat out in an even layer, going slightly up the sides.
Bake until the crust is no longer wet, 5 to 7 minutes. Remove from the oven.
Crack the eggs into the processor (there’s no need to clean it), add the vanilla and all but 1 tablespoon of the confectioners' sugar, and blitz until pale, about 2 minutes.
Buzz in the cream cheese and lemon zest and juice, then pour evenly over the cookie base.
Mash half the raspberries and the remaining tablespoon of confectioners' sugar with a fork, then swirl it through the top of the cream-cheese batter.
Slide the skillet in the oven and bake for 15 minutes.
Remove the skillet, scatter the remaining raspberries over the top of the cheesecake, and dust with more confectioners' sugar.
Bake the cheesecake until just set in the center, 15 to 20 minutes. Switch the oven to broil, and CAREFULLY broil until the top is golden in spots, 2 to 4 minutes more.
Remove the cheesecake from the oven and let it cool to room temperature, then chill in the fridge for 2 hours before serving.