Preheat the oven to 355°F/180°C (static). Line a 10x15-inch/25x38 cm jelly-roll pan with parchment and grease the paper.
In the bowl of a stand mixer or with a hand mixer, beat the eggs and sugar until pale and ribbon-like, about 3 to 4 minutes. Add the oil and vanilla and mix briefly to combine. Sift the flour, baking powder, and salt over the batter and fold gently until smooth and uniform.
Pour the batter into the prepared pan, spread to an even layer, and bake until the surface is just golden and springs back slightly, about 12–14 minutes. Remove from the oven, place a second sheet of parchment on top, invert onto a work surface, peel off the top paper, and roll the sponge up while still warm. Let the roll cool completely at room temperature, rolled.
While the sponge cools, whisk the oat- or cashew-based whipping cream, oat or cashew milk, instant pudding mix, sugar, and Biscoff Crunch spread until the mixture thickens to a light, stable cream. Chill briefly if it becomes too soft to spread.
Unroll the cooled sponge. Brush or spread the melted Biscoff Crunch (the 1/2 cup) thinly over the sponge and let it set for a minute. Spread the chilled Biscoff cream in an even layer, leaving a 1/2-inch border. Re-roll the cake tightly using the parchment to guide it, then wrap and freeze until firm, about 2 hours.
Set the frozen roll on a wire rack over a tray. Pour the melted smooth Biscoff spread over the frozen roll to coat. Drizzle with melted vegan white chocolate and scatter the ground Biscoff cookies on top. Chill the coated roll in the refrigerator for at least 2 hours to finish setting before slicing.
Store in an airtight container in the refrigerator for up to 3 days.