To make the dough: Mix the salt and flour in a medium bowl or a food processor. With a pastry blender, incorporate the butter until crumbs have formed. Add the water a few tablespoons at a time, until you can form a ball. Knead the dough for about 5 minutes. Let it rest for about 15 minutes more. You can prepare the dough in advance and refrigerate it.
To make the filling: Boil the potatoes until tender. Drain and set aside.
If using fresh carrots, chop and simmer in water, to cover, in a small pot. Add the corn and peas to barely cook. Set aside to cool.
Heat the oil in a pan and fry the onion until golden. Add the garlic, ginger, and chile and cook for 2 minutes. Add the garam masala, turmeric, chile powder, and salt and cook 2 minutes more.
In a bowl combine the mashed potatoes, the onion and spice mixture, carrots, peas, corn, lemon juice, and chopped coriander. Mix well.
To assemble the samosas: Divide the dough into 9 equal size balls. On a floured surface, roll each ball into a 5-inch circle. Cut each circle in half.
Brush the straightedge side with a little water, fold it in half, and align the two straight sides so they overlap to form a cone shape. Squeeze the edges together to make a tight seal. Place approximately 1 generous tablespoon of filling inside each cone, leaving the top edge clean. Moisten the inside top rim of the cone and press the edges together to make another tight seal. Place the samosas on a tray until ready to fry. Repeat with the remaining dough and filling.
Heat approximately 3 inches of vegetable oil in a deep saucepan. Fry several samosas at a time, being careful not to crowd them. When 1 side turns golden brown, flip it over to brown on the other side. Drain on paper towels. Serve with chutney.