Make the pastry by combining the flour and 1/4 teaspoon of the salt and then rubbing 1/4 cup of the ghee into the flour until well combined and mixture resembles coarse crumbs. Add the water, a tablespoon at a time, until the pastry comes together to form a ball, about 10 tablespoons. Knead the dough lightly and then form into a disk, wrap in plastic wrap and transfer to the refrigerator to rest for at least 1 hour.
While the dough is resting, make the filling. Peel the potatoes and cut into small dice. Heat 1/4 cup of ghee in a medium skillet and add the onion. Cook until the onion is lightly caramelized, about 6 minutes. Add the ginger, chile, coriander, garam masala, and cumin seeds and cook until the spices are fragrant, about 2 minutes. Add the potatoes, 1 teaspoon of the remaining salt, and 1/8 teaspoon of the pepper and cook, stirring, until well combined, about 6 minutes. Add the peas and 3 tablespoons of the cilantro and set aside to cool completely.
While the filling is cooking, make the yogurt dipping sauce by combining the yogurt, remaining tablespoon of cilantro, mint, green onion, lemon juice, remaining 1/4 teaspoon of salt and remaining 1/8 teaspoon of pepper and stir to combine. Set aside, refrigerated, until ready to serve the samosas.
Divide the samosa dough into 10 portions and roll each piece of dough into a circle about 5 inches in diameter. Cut each circle in half. Lightly moisten the straight edge of each half circle and press together to seal; you should now have a cone-shaped portion of dough. Carefully spoon about 2 heaping tablespoons of the filling into the dough cone and then moisten the remaining edges with a bit of water and press together to seal. Repeat with the remaining portions of dough and filling.
When you are ready to fry the samosas, heat a large saucepan filled at least 2 inches with oil to about 340 degrees F. Fry the samosas, a few at a time and stirring to promote even cooking, until golden brown, about 4 minutes. (Note: these will float to the surface of the oil long before they are ready to be removed; make certain to cook them until the pastry is crispy and golden brown.) Remove using a slotted spoon and transfer to a paper-lined plate to drain. Repeat until all samosas have been fried. Serve immediately, with the yogurt dipping sauce and your favorite chutney.