For the consommé, put the tomatoes in a mixing bowl with the tomato vines. Add the onion and basil to the bowl, season with salt and mix well to combine. Place the mixture in a food processor and pulse until broken down but still chunky. Transfer to a freezer bag, seal tightly and place in the freezer until frozen solid.
When ready to serve, defrost the mixture whilst still in the bag. Bash with a rolling pin until the mixture is well broken down. Line a colander with a large piece of muslin cloth and suspend it over a large bowl. Spoon the tomato mixture into the muslin and seal with pegs or string, then put the whole bowl into the fridge to drip away for at least 4 hours or overnight, without pressing or squeezing it at all.
Season with salt and pepper and add the vinegar and sugar. Chill until ready to serve. Any remaining tomato pulp can be used in a pasta or curry sauce.
Meanwhile, prepare the mille feuille. Preheat the oven to 190C/170C Fan/Gas 5 and line a baking tray with baking paper. Put the puff pastry on a lightly floured work surface and roll out to a thickness of 2mm. Cut the pastry into eight rectangles. Put the rectangles on the baking tray and chill in the fridge for 15 minutes. Place another baking tray on top and bake for 8 minutes. Remove the top tray and then bake again for another 3 minutes, or until golden brown and perfectly crisp. Leave to cool.
Put the burrata in a small bowl and season with salt and pepper. Add the oil and stir to mix.
For the pesto, put the basil, garlic, pine nuts and most of the olive oil in a food processor and blitz until smooth. Add more oil if preferred. Stir in the Parmesan and season with salt and pepper.
When ready to serve, assemble the mille feuille by placing the burrata pieces on one pastry rectangle. Top with another pastry rectangle and add more burrata pieces on top. Repeat with the remaining pastry rectangles and burrata until there are four mille feuille. Divide the diced tomatoes and basil leaves between four bowls and pour the consommé over the top into each bowl. Serve with the mille feuille and pesto on the side.