For the pastry, separate the eggs. Add 2-3 tbsp cold water to the egg yolks and stir with a fork. (You can use the egg whites for another dish or freeze them.)
Place the flour into the bowl of a food processor, followed by the cold butter and the salt.
Turn the processor on and pulse several times until the mixture resembles fine breadcrumbs.
Add two-thirds of the egg yolk/water mixture and pulse again. If the mixture is still too dry, add the remaining egg and water mixture (you may not need to use all of it). Be careful not to overwork the pastry. Stop pulsing when the mixture has the consistency of chunky breadcrumbs.
Turn the pastry out onto a clean, floured work surface and, with floured hands, bring together to make a smooth dough, but don't knead.
Shape into a flattened ball, wrap in cling film and refrigerate for about 10-15 minutes. You can keep the pastry at this stage for 2-3 days if not using it immediately.
Remove the pastry from the fridge. On a clean, floured work surface, roll it out with a floured rolling pin until it's slightly larger than the flan ring. Using the rolling pin, lift the pastry and lay it over the flan ring.
With your fingers, lightly press the pastry into the sides of the ring. Run a rolling pin over the top of the ring and pull away the excess pastry at the edges. Using your fingers, gently press the pastry up to slightly build up the height of the pastry at the edges. Prick the base of the pastry with a fork.
Make a cartouche (a circle of greaseproof paper) slightly larger than the base of the flan tin and line the pastry case with it. Place some ceramic baking beans inside to weigh the pastry down. Place in the fridge to relax for about 20 minutes.
Preheat the oven to 200C/400F/Gas 6. Place the pastry in the oven and bake blind for about 20 minutes. Remove the paper cartouche and baking beans for the final five minutes to crisp up the pastry.
Make the filling by whisking the eggs, egg yolk and sugar together until smooth and foaming, with a hand whisk or with an electric whisk. This will take 5-10 minutes.
Zest and juice the lemons and lime. Put the juice and zest into a bowl and mix together well. Whisk the cream into the egg and sugar mixture, then pour the lemon and lime juice and zest into the cream mixture. The acidity in the citrus will thicken the cream. Mix gently.
Using a jug, pour the filling mixture into the prepared pastry case, while it's still on the oven rack. Reduce the temperature to 150C/300F/Gas 2 and bake for about 50 minutes, or until just set.
Remove the tart from the oven and allow to cool completely before chilling in the fridge. Carefully remove the flan ring. Dust with icing sugar just before serving.
Serve with fresh raspberries and thick cream.