Preheat oven to 350°F and line jelly roll pan with parchment paper.
Beat eggs and sugar on high for 5–7 minutes until thick and pale.
Gently fold in vanilla, then sift in flour, baking powder, and salt.
Add milk and mix gently until just combined.
Spread batter evenly and bake 12–14 minutes.
Dust a towel with powdered sugar and flip hot cake onto it.
Peel parchment, roll cake in towel, and cool 1 hour.
Beat cream cheese, sugar, and vanilla until smooth.
Whip cream to stiff peaks and fold into cream cheese.
Swirl in strawberry puree and jam.
Unroll cake, spread filling, add strawberry slices.
Roll up again, wrap, and chill 2+ hours.
Dust with powdered sugar and glaze before serving.