Pulse 14.5 ounces tomatoes, with their juice, in a food processor until mostly smooth, about 8 pulses.
Heat 2 Tablespoons oil in a large Dutch oven over medium-high heat until shimmering. Add 3 teaspoons mile or sweet curry, 1 teaspoon red curry, powders and 1 teaspoon garam masala, and cook until fragrant, about 10 seconds. Add 2 minced onions, ½ cut yam and 1 cut potato and cook, stirring occasionally, until the onions are browned and the vegetables are softened slightly, about 7 minutes. Reduce the heat to medium. Clear the center of the pot, stir in the remaining 1 Tablespoon oil, 3 minced garlic, 1 Tablespoon ginger, 1 Tablespoon tomato paste, and 1 minced chile, and cook about 30 seconds. Add ½ cauliflower and cook, stirring occasionally, until the spices coat the florets, about 2 minutes.
Stir in the processed tomatoes, 1 ¼ cup water, 14.5 ounces chickpeas, and ½ teaspoon salt, scraping up any browned bits. Bring to a simmer, cover and cook, stirring occasionally, until the vegetables are tender, about 15 minutes.
Stir in 1 ½ cups peas and continue to cook until heated through, about 2 minutes longer. Off the heat, stir in ½ cup yogurt. Serve over rice with ½ chopped cilantro, cashews and green onions for garnish.