Preheat oven to 400°F (200°C) and line a 15x10-inch rimmed baking sheet (or jelly roll pan) with parchment paper.
In the bowl of a stand mixer fitted with the whisk attachment, beat eggs, sugar, and vanilla on medium-high speed until thick and pale and tripled in volume, about 8–10 minutes.
In a small bowl, combine the flour, salt, and baking powder.
In three additions, sift the flour mixture into the egg mixture, using a silicone spatula to fold it in after each addition.
Transfer batter to the prepared baking sheet. Smooth the top with an offset spatula. Bake until light golden brown, about 8–10 minutes.
Dust a clean kitchen towel generously with powdered sugar and invert the baking sheet with the cake onto it; slowly remove the parchment paper. You can brush the paper with a little bit of water to make it easier to peel off.
Using the towel and starting at a short end, roll up the cake and towel. Let cool completely, rolled into the towel, seam side down.
When the cake has cooled, make the strawberry filling.
Using a handheld immersion blender or food processor, purée the strawberries. You should have about 1/2 cup strawberry purée.
In a small saucepan, reduce the purée by half. Let cool.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter and powdered sugar at medium speed until pale and fluffy.
Add cream cheese and mix until the mixture is whipped, white and fluffy, about 1–2 minutes. Scrape down the bowl as needed.
Add the strawberry purée and stir into the mixture on low speed until just fully incorporated.
Carefully unroll the cake. Spread the strawberry filling evenly over the surface, leaving a 1/2-inch border on the outer short end. Gently roll up the cake tightly in the same direction you rolled it the first time (without the towel).
Place it on a serving platter and cut off the ends for a prettier cake if desired.
To garnish, make whipped cream by whipping 1/2 cup heavy cream with 1 tbsp powdered sugar on medium-high speed until soft peaks form.
Garnish the cake with dollops of whipped cream and strawberries.