Zest both lemons. Set zest aside. Juice lemons until you have 1/4 cup juice. Mix juice with 1/4 cup sugar, and 2 sprigs thyme in a small saucepan. Bring to a boil, until the sugar is fully dissolved. Set aside to cool.

In a small bowl, stir together gelatin with water and set aside to soften. In a large saucepan, combine milk, reserved lemon zest, and 3 sprigs of thyme. Place over medium heat. While milk is heating, whisk together yolks and remaining 1/3 cup sugar until lightened in color, about 2 minutes. When milk is steaming but not yet boiling, remove thyme and slowly whisk in half of the milk mixture into yolk mixture. Once combined, whisk in remaining milk before returning to saucepan over medium heat. Whisk in lemon curd and bloomed gelatin. Continue whisking until gelatin is fully dissolved and mixture has thickened to coat the back of a spoon. Strain into a large bowl to cool to room temperature, about 1 hour and 15 minutes, stirring occasionally.

While custard cools, line the side of a 9-inch springform pan with a strip of parchment paper or acetate wide enough to come up over the edge. Take 24 of the lady finger halves and cut off the rounded edge of one end of the lady fingers so that they’re flat and make a 90° angle with the long side of the lady finger. Line the side of the pan with these lady fingers by standing them side-by-side and up straight so the cut flat bottoms are flush with the bottom of the pan. Once the perimeter is tightly packed, line the bottom with the remaining lady fingers, pressing them close together so there are no spaces in the base. Remove fresh thyme from reserved lemon-thyme syrup, and brush the lady fingers evenly with all of the syrup. Evenly spread blackberry jam across the base of the cake. Cut half of the blackberries in half lengthwise and place them face down on top of the blackberry jam.

Once custard has cooled, whip 1 cup cream to medium peaks. Gently fold in 1/3 of the custard into the cream. Once well incorporated, fold the custard mixture into the remaining whipped cream. Pour into prepared pan, using a spatula to level the mixture inside the pan. Refrigerate, covered with plastic wrap, for 4 hours (or overnight) until set.

Whip remaining 1/2 cup cream with powdered sugar. Remove the side of the springform pan and parchment paper. Surround the perimeter of the top of the cake with remaining blackberries. Dust with powdered sugar. Fill the rest of the center by decoratively piping the whipped cream. Finish with a sprinkling of thyme leaves.