Prepare the Apricot Jelly: Blend the apricots into a puree. Heat the puree with 2 cups of sugar, lemon juice, and agar-agar. Bring to a boil and simmer for a few minutes. Add the pre-soaked gelatin sheet, mix well, and let it cool.
Line a tray with plastic wrap, pour the apricot jelly evenly to form a uniform layer, and chill for 2 hours.
Preheat the oven to 356 °F (180 °C).
Whisk the egg yolks with 1/4 cup of sugar until pale. Sift together the flour, cornstarch, almond flour, and baking powder, then add to the egg-sugar mixture. Stir in the almond extract.
Beat the egg whites with the remaining 1/4 cup of sugar until soft peaks form (like Italian meringue). Gently fold this meringue into the previous batter.
Line a baking sheet with parchment paper. Pour the batter onto the sheet, spread it evenly, and bake for 10 minutes.
Once baked, cover the sponge cake with a damp cloth and let it cool slightly. Turn the cake onto the apricot jelly layer, remove the parchment paper, and roll it up carefully using the plastic wrap. Tighten the wrap and chill for 2 hours.
Before serving, sprinkle with toasted sliced almonds.