Preheat the oven to 180°C. Finely chop the shallots and sauté them for 5 minutes on low heat in a saucepan with 2 tbsp of oil. Add the vinegar and let it reduce completely. Let it cool down.
Mix the Philadelphia with the chopped chives, salt, and pepper. Slice the beetroot into rounds.
Cut the puff pastry into 4 and slightly cut them to make them rounder. Place a generous tablespoon of cheese in the center of each sheet, add shallots, and form a mound. Arrange slices of beetroot around it and brush with olive oil. Fold the edges of the pastry inward and brush with beaten egg. Bake for 35 minutes until golden brown in the oven.
Mix the rapeseed oil, grated lemon zest, and lemon juice. Peel the Jerusalem artichokes and cut them into fine brunoise. Mix them with the dressing and add the pomegranate seeds; season with salt and pepper.
Chop the mint and pistachios.
Sprinkle the tarts with mint and pistachios and serve them very warm with the salsa.