Heat the oven to 175 C / 350 F with a rack in the center. Line a 25 x 38 cm / 10 x 15-inch jelly roll pan with parchment, pressing it flat and leaving a 2 cm overhang on the short ends.

Whisk the egg yolks with 40 g of the sugar for about 2 minutes until pale. Whisk in the milk, oil, 60 g halaya, ube extract, and salt until no lumps of halaya remain, then sift the cake flour over the bowl and fold until no dry streaks show.

In a clean, dry bowl, beat the egg whites with the cream of tartar on medium speed until foamy. Add the remaining 60 g sugar a tablespoon at a time, then beat on high for 3 to 4 minutes until the peaks are stiff and glossy and stand straight when you lift the beaters. Fold one-third of the whites into the yolk batter to loosen it, then fold in the rest in two additions until no white streaks remain.

Spread the batter into the pan in an even layer and tap the pan twice on the counter to release large bubbles. Bake for 12 to 14 minutes, until the top springs back under a fingertip and the edges barely pull from the parchment. The sponge won't brown much, so go by the spring-back and not by color.

Run a knife around the edges, flip the sponge onto a fresh sheet of parchment, and peel off the baking parchment. Starting at a short end, roll the warm sponge up with the parchment inside, then rest it seam side down on the counter for 20 minutes.

While the sponge cools, beat the cold cream, powdered sugar, and vanilla to soft peaks. Add the 100 g loosened halaya in three additions, beating after each, until the cream is stiff enough to hold a spoon-trail shape.

Unroll the cooled sponge and spread the filling over it, stopping 2 cm from the far edge. Re-roll tightly, using the parchment to guide the first turn, then wrap the roll in the parchment and chill it seam side down for at least 30 minutes.

Trim 1 cm off each end with a serrated knife, then cut the roll into 8 slices about 3 cm wide, wiping the blade clean between cuts.