For the dough: Whisk together the flour and 1 teaspoon salt in a large bowl. Drizzle the butter over the flour. Rub the butter into the flour between the palms of your hands until evenly distributed. Make a well in the middle of the mixture. Pour 1/2 cup plus 1 tablespoon water into the well. With a fork, gradually stir the flour into the water to form a dough. It should be firm and not sticky. If it is too dry add another tablespoon of water. Knead until smooth, then return the dough to the bowl and cover with plastic wrap. Let it rest at room temperature for 30 minutes.
For the filling: Put the potatoes, 2 tablespoons salt and 2 quarts of water in a large pot. Bring to a boil over medium-high heat, then reduce to a simmer and cook until halfway cooked, 12 minutes. Add the cauliflower florets and simmer until the potatoes and cauliflower are fully tender, about 5 minutes more. Drain well and let the vegetables cool completely.
Heat a large skillet over medium-high heat. Before the pan gets fully heated, add the butter. When it starts to turn nut brown, after about 2 minutes, add the onion, cumin seeds and 1/4 teaspoon salt. Cook, stirring frequently, until the onion is golden brown, about 5 minutes. Add the garlic, ginger and chile. Cook for 1 minute, then sprinkle in the garam masala and turmeric. Stir for another 20 seconds, then add the potato and cauliflower and 3/4 teaspoon salt. Cook 3 minutes while tossing continuously. Add about 1/3 cup water and scrape up any browned bits stuck to the pan. Cook for an additional 2 minutes. At this point, the mixture should be dry and contain a combination of whole and mashed pieces. (This texture will help the filling hold together.) Take the pan off the heat and fold in the peas, lime juice and sugar. Let cool completely.
To make the samosas: Divide the dough into 8 equal balls. Roll out each to a 7-inch disc. If the dough sticks a little, dust the surface lightly with flour. Cut each disc in half to create 16 semi-circles. Keep any dough you are not working with covered with plastic wrap or a clean dishtowel. Working with one piece of dough at a time, bring the two corners of the semi-circle together so they overlap by about an inch to form a cone. Brush the egg where the dough overlaps and pinch shut with your fingers. Holding the dough in your hand like an ice cream cone, fill the cone with about 2 1/2 tablespoons of the filling. Seal the top of the cone by brushing one edge with the egg wash and folding the other edge over it; pinch to seal. If there is a hole at the bottom of the cone, pinch it shut while maintaining a nice pointy shape. Turn the samosa onto the work surface seam-side down so it flattens and creates a base. Repeat with the remaining dough and filling.
Line a baking sheet with several layers of paper towels. Heat 3 inches of oil in a heavy-bottomed pot or Dutch oven to 350 degrees F.
Place 8 samosas in the hot oil and cook until golden brown, about 9 minutes, adjusting the heat level if necessary in order to maintain a temperature of 325 degrees F. Drain well on the prepared baking sheet. Bring the temperature of the oil back up to 350 degrees F and repeat with the remaining 8 samosas.
Serve with tamarind or mango chutney.
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