Preheat the oven to 350. Line a rimmed baking sheet with parchment and spray with cooking spray.
In a large bowl, whisk together the eggs until they are light yellow. Add the sugar, oil, baking powder, salt, and vanilla, and mix until well combined. Add the flour, and mix until just combined.
Pour the batter into the prepared baking sheet, spreading evenly. Bake 10-15 minutes, until lightly browned and a toothpick inserted in the middle comes out clean.
Spread out a clean kitchen towel and sprinkle with powdered sugar. Invert the cake onto the towel and remove the parchment sheet. Fold the edge of the towel over the short end of the cake and roll up into a spiral, with the towel rolled inside the cake. Refrigerate until cooled completely.
While the cake is cooling, beat together the cream cheese and sugar. Add the whipping cream and vanilla, and beat until light and fluffy.
Unroll the cooled cake and spread the cream cheese filling over the top of the cake. Spread the strawberries in an even layer over the cake. Roll the cake up again, but not so tightly that the filling squishes out. Place seam-side down on a serving plate. Sprinkle the top of the cake with powdered sugar and refrigerate 30 minutes before slicing and serving.