Preheat the oven to 350°F/180°C, and line two baking sheets with parchment paper. If you have only one baking sheet, that's also fine, you can reuse the same one once the first set of lady fingers is done.
In a medium bowl, sift together the flour, cornstarch, salt, and baking powder. Stir to combine and set aside.
Pour the aquafaba and the cream of tartar in the bowl of a very clean and wiped dry stand mixer and whip on high speed with the wire whip attachment until stiff peaks are formed. Start on medium for a few seconds, then move up to high speed. This should take about 2-3 minutes to get stiff peaks.
Reduce the stand mixer speed to low, and slowly add the powdered sugar. When all the powdered sugar is added in, turn the speed back to high, and whip until the mixture is shiny. It may not be completely stiff but in nice ribbons.
Reduce the mixer to low and slowly add the vanilla and butter, mixing until well combined. Increase speed back to high for about a minute until everything is combined, then turn off the stand mixer.
Remove the bowl from the stand mixer and add the sifted flour mixture in three additions, folding gently with a rubber spatula until just combined and no flour pockets remain, being careful not to deflate the aquafaba.
Transfer the batter to a zip-top bag with a corner cut off or a piping bag with a large round tip (I cut a wide corner to get that inch out). Pipe the mixture into thick lines about 3 inches long and an inch wide on the prepared baking sheets, leaving at least an inch between each cookie.
Bake for 15 minutes, until the edges are golden and the cookies are set in the center. Remove from the oven and cool for 5 minutes on the baking sheets, then transfer to a wire rack to cool completely.
Notes
Room temperature: Let the ladyfingers cool completely, then place them in an airtight container with parchment paper between the layers. Store in a cool, dry place for 2 to 3 days.
Freezer: Freeze the ladyfingers on a parchment-lined baking sheet, then place them in an airtight container or freezer bag and freeze for up to 2 months. Thaw at room temperature before serving.