Preheat oven to 200°C (400°F), with a rack in the center of the oven. Place berries, vanilla extract and 50g (1/4 cup) of sugar into a medium baking dish (I use a pie dish). Toss to combine, then bake in the oven for 20 minutes, until the berries are bubbling and bursting. Remove from oven and set aside to cool.
Prepare the pastry as the berries are roasting. Place leftover sugar in a small bowl and mix in the lime zest. Line a large baking sheet with parchment paper, then place one phyllo sheet on it. Brush with butter and sprinkle with a heaping tablespoon of lime sugar. Top with another phyllo sheet and continue in the same manner until you’ve stacked 6 sheets, brushing the last sheet with butter and sugar too. Bake in the preheated oven until crisp and golden, 10-12 minutes. As this bakes, prepare the second phyllo stack the same way you did the first.
When the first phyllo stack is baked, remove it from the oven and put the second one in. Then line a 9x13-inch (23x33-cm) pan with plastic wrap. Measure and trim the warm phyllo stack, so that it fits into the 9x13 pan (you’ll probably have to cut about an inch from each side). Transfer the trimmed phyllo stack into the pan. When baked, trim the second phyllo stack too and set it aside.
When you’re ready to make the sandwiches, place ice cream in your stand mixer or food processor. Beat until it’s soft and spreadable, but still icy. Quickly stir in the roasted berries, then spoon the ice cream into the prepared pan (with one phyllo stack on the bottom). Spread ice cream evenly all over, top with the second phyllo stack, wrap well (I use both plastic wrap and aluminium foil) and freeze until solid. This will take at least 4-6 hours, but it can easily wait overnight.
When ready to serve, cut the ice cream into 20 equal pieces (or how many you want). Serve how many you want, put the rest back into the freezer immediately.