Prepare the dough: In a large bowl, combine all-purpose flour, ghee (or oil), carom seeds, and a pinch of salt. Mix well. Gradually add water, kneading to form a firm, smooth dough. Cover the dough with a damp cloth and let it rest for at least 30 minutes.

Prepare the potato base filling: Heat 1 tablespoon of oil in a pan over medium heat. Add cumin seeds and let them splutter. Add ginger-garlic paste and sauté for 30 seconds until fragrant. Add the mashed potatoes, green peas, turmeric powder, garam masala, red chili powder, and salt. Mix everything well and cook for 5-7 minutes, breaking up any lumps, until the spices are well incorporated. Stir in the fresh cilantro. Let the filling cool completely.

For Mixed Vegetable Samosas: Divide the cooled potato base filling. To one portion, add the finely diced carrots, green beans, and corn kernels. Mix thoroughly to combine.

For Cheese Samosas: To another portion of the cooled potato base filling, add the grated cheddar cheese. Mix until the cheese is evenly distributed.

Assemble samosas: Divide the rested dough into 8-10 equal balls. Roll each ball into an oval shape, about 6-7 inches long. Cut the oval in half to create two semicircles. Take one semicircle, moisten the straight edge with water. Bring the two ends of the straight edge together to form a cone, sealing them securely. Fill the cone with your desired filling (either the mixed vegetable or cheese potato mixture). Moisten the top edges of the cone and gently press them together to seal the samosa into a triangle shape. Ensure all edges are tightly sealed to prevent filling from coming out during frying.

Fry samosas: Heat enough oil for deep frying in a heavy-bottomed pan or wok over medium-low heat. Once the oil is hot (around 325-350°F / 160-175°C), carefully slide the assembled samosas into the oil, without overcrowding the pan. Fry the samosas on medium-low heat, turning occasionally, until they are golden brown and crispy on all sides (this will take about 8-12 minutes per batch). The low heat ensures the dough cooks through and becomes crispy without burning.

Remove the fried samosas with a slotted spoon and place them on a plate lined with paper towels to drain excess oil.

Serve hot: Serve the delicious homemade Mixed Vegetable Samosas and Cheese Samosas immediately with your favorite chutneys.