In a medium saucepan, cover the pigeon peas with water and let soak for 30 minutes. Drain the pigeon peas well.
Return the pigeon peas to the saucepan. Add the halved jalapeño, 1/2 teaspoon of the turmeric, 4 cups of water and a pinch of salt and bring to a boil. Simmer over moderately low heat until the peas are just tender, but not falling apart, about 15 minutes. Discard the jalapeño halves.
Meanwhile, in another medium saucepan, heat the canola oil until shimmering. Add the cumin seeds and cook over moderate heat until fragrant, about 1 minute. Add the onion, garlic, ginger and minced jalapeño and cook over moderate heat, stirring, until the onion is lightly browned, about 5 minutes. Add the remaining 1/2 teaspoon of turmeric along with the tomatoes and cayenne and cook over moderately low heat until the liquid has evaporated and the tomatoes are softened, about 5 minutes longer.
Add the pigeon peas and their cooking liquid to the tomato mixture and simmer until the dal is slightly thickened, about 5 minutes. Season with salt. Stir in the butter and cilantro and serve with rice and warm naan.