Place the dal in a medium pot, cover with cold water, and let sit overnight or for 8 hours. Drain and return to the pot with chile halves and 3 cups water. Season with salt and bring to a boil. Reduce the heat to medium-low and cook, covered, until just tender but not mushy, about 15 minutes. Remove from heat and discard chile.
In a large skillet, heat ¼ cup oil on medium-high. Add onion, garlic, minced chile, ginger, cumin, turmeric, mustard seeds, and curry leaves, and cook, stirring occasionally, until onions are soft and just beginning to brown. Add tomato, season with salt, and cook until softened.
Add mixture to dal and return to a simmer. Cover and cook on low heat until very tender, about 15 minutes. Remove cover and cook over medium-high heat until liquid is slightly reduced. Season to taste with salt. Using a whisk, add butter and whisk to break up some of the dal.
To serve, divide into bowls with rice and top with cilantro. Serve with a side of naan.