Make the dough: Pulse flour, butter, and salt in a food processor into pea-size crumbles. Add water; pulse until dough forms. Divide into 12 balls; chill 1 hour.

Make the filling: Boil potatoes and carrots in a 4-qt. saucepan of salted water until tender, 8–10 minutes. Drain; coarsely mash. Add 2 tbsp. oil to pan; heat over medium-high. Cook cumin seeds until they pop, 1–2 minutes. Add onion and ginger; cook until golden, 4–6 minutes. Let cool; stir into potato mixture with peas, cilantro, mint, garam masala, chiles, and salt.

Form and fry samosas: Working with 1 ball at a time, roll dough into a 6" round; cut in half. Gather straight edges of 1 half-round together, overlapping by 1⁄4" to form a cone. Moisten seam with water; press to seal. Spoon 1 tbsp. filling into cone. Moisten edges of cone with water; pinch to seal.

Heat 2" oil in a 6-qt. saucepan until a deep-fry thermometer reads 180° C 350° F. Fry samosas until crisp, 8–10 minutes. Drain on paper towels; serve with whatever chutneys if you like.