Ppreheat the oven to 180℃/160℃ fan/350℉/gas mark 4. Line two baking sheets with baking parchment.
Sift together the plain flour, semolina, cornstarch and baking powder well. Set aside.
Place the aquafaba and cream of tartar in a large, spotlessly clean bowl or the bowl of a stand mixer fitted with the whisk attachment. Beat with an electric whisk until it forms soft peaks.
Add the caster sugar a spoonful at a time, whisking well after each addition. Continue to whisk until the meringue is thick and glossy and forms stiff peaks.
Add half of the dry ingredients and fold through very gently with a spatula until mostly combined, it doesn't matter if there are still some dry patches. The meringue will deflate a bit, this is normal.
Add the oil and vanilla extract and fold through a couple of times, it does not need to be fully combined.
Add the other half of the dry ingredients and gently fold in until just combined, with no dry patches; do not over-mix. The batter should be thick, sticky and full of air bubbles.
Transfer the batter to a large piping bag fitted with a large round nozzle. Alternatively you can just snip the end off of a disposeable one.
Pipe lines of the batter well spaced apart on the baking sheets, about 2.5 cm (1 in) wide and 8-10 cm (3-4 in) long.
Sprinkle with caster sugar then generously dust with icing sugar using a fine mesh sieve.
Bake for 15-20 minutes until golden. This will produce ladyfingers that are crisp on the outside and soft on the inside. For drier, crunchier cookies, recuce the oven temperature to 150℃/130℃ fan/300℉/gas mark 2 and bake for a further 10-15 minutes.
Carefully transfer the ladyfingers to a wire cooling rack and leave to cool completely. Once cool, store in an airtight container.