Place potatoes in a small bowl and cover with cold water. Place split peas in a large saucepan. Fill the pan halfway with water and rinse the peas by rubbing them between your fingers. (The water will become cloudy.) Drain. Repeat three or four times, until the water remains relatively clear; drain. Add 4 cups water to the split peas and bring to a boil over medium-high heat. Skim off any foam that rises to the surface. Drain the potatoes and add to the peas. Return to a boil, reduce heat to medium and simmer, uncovered, for 5 minutes.

Stir in cauliflower, green beans, eggplant, carrot, salt and turmeric. Return to a boil; cover, reduce to a gentle simmer and cook, stirring occasionally, until the vegetables are fork-tender and the peas are soft but firm-looking, 7 to 10 minutes more.

Meanwhile, heat oil in a small skillet over medium-high heat. Add cumin seeds and cook until they sizzle and smell fragrant, 15 to 20 seconds. Stir in garlic and chiles to taste and cook, stirring, until the garlic is light brown and the chiles are fragrant, 1 to 2 minutes. Remove from the heat.

Stir the garlic-chile mixture into the cooked vegetables. Scoop a ladleful of cooking water from the saucepan to the skillet; swish it around and pour the "washings" back into the saucepan.

Whisk cornstarch with 3 tablespoons of the cooking liquid in a small bowl until smooth. Stir it into the stew along with cilantro and ginger. Increase heat to medium-high and simmer the curry, uncovered, stirring occasionally, until the sauce thickens, about 2 minutes. Stir in lime juice and ghee (or butter), if using.