Let the ice cream soften for a few minutes, then stir the chopped strawberries into it. Place on a piece of parchment or waxed paper and form into a log as long as the length of your jelly roll pan. Wrap and put in the freezer.
Preheat the oven to 350 degrees. Line a jelly roll pan with parchment paper and butter the paper or spray with cooking spray.
In an electric mixer fitted with the whisk attachment, beat the eggs and sugar on high speed until very pale yellow and doubled in volume, about 2 minutes. Add the flour, baking powder, and salt. Beat until fully incorporated. Beat in the vanilla. Spread the batter evenly on the prepared pan – it will be thin. Bake for 11-13 minutes or until set in the middle. Remove from the oven and let cool for 2 minutes.
Using the parchment paper, roll the cake into a cylinder, starting at one of the short ends. Let cool for 20 minutes.
Gently unroll the cake and spread evenly with the jam. Place the ice cream log at one end and roll the cake up. Wrap back in the parchment and freeze until serving.
Unwrap the roll, place on a serving plate, and dust with powdered sugar. Garnish with strawberries, mint, and chopped pistachios.