Preheat oven to 375°F (190°C). Line a 10×15-inch jelly roll pan (or similar rimmed baking sheet) with parchment, leaving some overhang on the long sides. Lightly spray or brush the parchment with nonstick spray or melted butter. Lay a clean, thin kitchen towel on the counter and dust it generously with powdered sugar; this is where the hot cake will be inverted.

In a large mixing bowl or stand mixer, combine the 4 room temperature eggs and granulated sugar. Beat on medium-high speed for 5–7 minutes, until the mixture is very pale, thick, and at least doubled in volume and falls in a thick ribbon from the beaters. Gently mix in the milk and 1 teaspoon vanilla just until combined.

In a separate bowl, whisk together the flour, baking powder, and salt. Sift or lightly sprinkle the dry mixture over the egg mixture in 2–3 additions, folding gently with a spatula after each addition. Use broad, gentle folding motions from the bottom of the bowl and stop as soon as no dry streaks remain to avoid deflating the batter.

Pour the batter into the prepared pan and spread it gently and evenly into the corners, trying not to tap the pan. Bake for 9–12 minutes, until the top is lightly golden and springs back when gently pressed in the center; a toothpick should come out clean or with a few tiny crumbs. Do not overbake.

Immediately after removing the cake from the oven, run a knife around the edges if needed to loosen. Carefully invert the hot cake onto the powdered sugar–dusted towel. Peel off the parchment paper. Starting from a short end, gently roll the warm cake up with the towel into a log (not too tight, just snug). Place the rolled cake, seam-side down, on a cooling rack and let it cool completely, about 45–60 minutes.

Once the cake is cool, add the cold heavy whipping cream, 3 tablespoons powdered sugar, and 1 teaspoon vanilla to a chilled mixing bowl. Beat on medium-high speed until medium-stiff peaks form; the cream should be thick and spreadable but not grainy. Adjust sweetness to taste if desired.

Carefully unroll the cooled cake; it will retain some curve from being rolled. Stir the jam until smooth, then spread a thin, even layer over the surface of the cake, leaving about a 1/2-inch border on all sides. Dollop the whipped cream over the jam and spread it evenly, again leaving a small border to prevent overflow. Using the towel to help, roll the cake back up from the same short side, this time without the towel inside. Place the filled roll seam-side down on a serving platter.

Wrap the swiss roll loosely in plastic wrap or cover with a cake dome. Chill for at least 1–2 hours to help the filling firm up. Just before serving, dust the top with additional powdered sugar. Slice with a sharp serrated knife, wiping the blade between cuts for neat spirals, and serve chilled or at cool room temperature.