Boil the potatoes, unpeeled, in salted water until a fork slides through easily, 20 to 25 minutes. Drain, let them steam dry for 5 minutes, then peel and mash coarsely.
Heat the oil in a skillet over medium heat. Add the mustard seeds and wait until they pop, then add the curry leaves, hing and ginger and stir for 30 seconds.
Stir in the turmeric, potatoes and salt until evenly yellow. Take the pan off the heat, stir in the lemon juice and cilantro, and cool completely.
For the green chutney
Blend the cilantro, chili, ginger, lemon juice and salt with just enough water to make a thick paste.
For the peanut garlic chutney
Toast the sesame seeds in a dry pan until golden, then cool.
Pulse the peanuts, sesame, garlic powder, chili powder and salt into coarse crumbs.
For the braid
Heat the oven to 400°F (200°C) and line a sheet pan with parchment paper.
Lay the pastry on the parchment. Spread 3 tablespoons of green chutney down the middle third and sprinkle 3 tablespoons of peanut garlic chutney over it.
Pile the cold potato filling on top and shape it into a neat log.
Cut 1-inch diagonal strips down both sides, from the edge in to the filling. Fold the strips over the filling, alternating left and right, and tuck the ends under.
Brush with the beaten egg.
Bake for 25 to 30 minutes, until deep golden and browned underneath.
Rest for 10 minutes, then slice across and serve with the remaining chutneys.
Notes
Contains peanuts. Label it if it's going to a party.
Leave the seeds in the chutney's green chili for more heat.
Make the filling and both chutneys a day ahead and keep them in the fridge.
Fried fresh garlic in place of garlic powder gives a sharper chutney.
Store leftovers in the fridge up to 3 days. Re-crisp at 350°F for 8 to 10 minutes.