Preheat the oven to 350 °F. Line a 10 by 15 inch jelly roll pan with parchment paper and lightly grease the parchment with oil or melted butter.
In a large mixing bowl, beat the eggs and granulated sugar with a hand mixer or stand mixer on medium high speed for 5 to 7 minutes until very pale, thick, and tripled in volume. The mixture should fall from the whisk in a slow ribbon.
Beat in the vanilla bean paste just until combined.
In a separate bowl, whisk together the flour, baking powder, and salt.
Sift half of the dry mixture over the whipped eggs. Gently fold with a spatula using broad strokes, then sift over the remaining dry ingredients and fold again just until no dry streaks remain.
Sprinkle the lemon zest over the batter, drizzle in the oil or melted butter around the edge of the bowl, and gently fold until evenly incorporated without deflating the batter.
Pour the batter into the prepared pan and spread it into an even layer, reaching the corners. Tap the pan once on the counter to release large air bubbles.
Bake for 10 to 14 minutes, until the top is light golden, the cake springs back when lightly pressed, and the edges just begin to pull away from the sides of the pan.
While the cake bakes, lay a clean, lint free kitchen towel on the counter and dust it generously with powdered sugar.
Immediately after removing the cake from the oven, run a thin knife around the edges, then invert the hot cake onto the sugared towel. Carefully peel off the parchment paper.
Starting from a short side, use the towel to help roll the warm cake into a snug spiral, with the towel rolled inside. Place the rolled cake seam side down on a wire rack and let it cool completely.
While the cake cools, make the filling. In a medium bowl, beat the cream cheese, powdered sugar, lemon zest, and vanilla until smooth and creamy.
In a separate bowl, whip the cold heavy cream to soft medium peaks. Gently fold the whipped cream into the cream cheese mixture until smooth and fluffy.
Once the cake is completely cool, carefully unroll it. It is normal if it curves slightly and does not lie completely flat.
Spread the lemon cream filling evenly over the cake, leaving a 1/2 inch border around the edges to prevent the filling from squeezing out.
Roll the cake back up from the same short side, this time without the towel, keeping the roll snug but not overly tight.
Wrap the filled roll firmly in plastic wrap and chill in the refrigerator for at least 1 hour to set the filling and firm the cake.
Before serving, unwrap the roll, trim the ends for a neat look if desired, slice into thick rounds, and serve chilled or at cool room temperature.