Preheat to 350°F (175°C). Line a jelly roll pan with parchment. Mix cake batter with eggs, oil, water, and vanilla. Spread and bake for 12–15 minutes.
Dust a towel with powdered sugar. Invert hot cake onto it, peel parchment, and roll it up with the towel. Cool completely.
Beat cream cheese and butter until smooth. Add powdered sugar and vanilla, then fold in crushed Oreos.
Unroll the cake, spread filling evenly, and leave a small border.
Roll it back up (no towel), wrap in plastic, and chill for 1 hour.
Drizzle with melted white chocolate and top with whole and crushed Oreos.
Slice into 1-inch pieces and serve chilled or right away.