Wash, hull, and finely chop the strawberries, then toss them with 2 tablespoons of sugar and set aside to release their juices and deepen the flavor.
Preheat the oven to 350°F (175°C) and line a 10×15-inch jelly roll pan with parchment paper, lightly greased. Beat the eggs and sugar for 5–7 minutes until pale and thick, then add vanilla and mix gently.
Sift flour, baking powder, and salt, folding into the egg mixture in two additions. Combine milk and vegetable oil, fold into the batter until smooth, pour into the pan, and bake 10–12 minutes until golden and springy.
While still warm, invert the cake onto a powdered sugar-dusted towel and peel off the parchment. Roll the cake in the towel from the short end and let it cool completely to prevent cracks.
Beat the cream cheese until smooth, mix in powdered sugar and vanilla, then fold in whipped heavy cream until light and fluffy.
Unroll the cooled cake, spread the cheesecake filling evenly leaving a small border, and scatter the macerated strawberries over the top. Carefully roll the cake back up without the towel, wrap in plastic, and chill 1–2 hours.
Dust with powdered sugar, garnish with fresh strawberries or whipped cream, slice into 1-inch pieces, and serve chilled.