Make samosas: Simmer potato and cauliflower in water to cover until just tender, 5 to 7 minutes. Drain and chop.
Heat oil in a large frying pan over medium heat. Add mustard seeds, cover, and cook until they pop, 30 to 60 seconds. Stir in turmeric and cayenne and cook just until sizzling. Add onion, garlic, and ginger and cook, stirring often, until softened, about 3 minutes.
Remove pan from heat and stir in garam masala, salt, lemon juice, and potato mixture. Stir in cilantro.
Preheat oven to 425°. Lay pastry flat on a work surface and cut each round into 8 pie-shaped wedges. Mound about 1 1/2 tbsp. potato filling on center of each wedge. To shape each samosa, brush edges of a pastry wedge with water, then pull up points to meet in center. Pinch points and edges to seal, neatening points. Transfer samosas to a rimmed baking sheet.
Bake samosas until golden, 18 to 20 minutes. Meanwhile, make mint chutney: Whirl mint, chile, onion, sugar, salt, lemon juice, and 3 tbsp. water in a blender until smooth. Pour into a small bowl. Spoon tamarind chutney into another bowl. Serve samosas hot with chutneys.