Make the bhaji: Heat the vegetable oil and butter in a large pot or dutch oven over medium heat. Add the onion and cook until soft and golden, about 20 minutes, reducing the heat if it browns too quickly.
Add the mustard seeds and curry leaves and sauté 1–2 minutes until fragrant (be cautious, as the mustard seeds may pop and sputter). Add the ginger, serrano peppers and tomatoes and cook 1 minute. Add the turmeric, cayenne and salt and cook until the oil glistens on top, 3–5 minutes.
Add the potatoes and carrots, then top with 3 cups of water. Cover the pot and bring to a boil over high heat, then reduce to medium and cook until the potatoes and carrots are very soft, about 30 minutes, stirring often. Coarsely mash with a potato masher or wooden spoon, then stir in the cilantro and lime juice.
Make the pooris: In a large bowl (or stand mixer), combine the flour, salt and baking powder. Add the warm water and knead until the dough comes together, adding more water or flour as needed for a workable, not-too-sticky dough. Knead about 5 minutes, then let the dough rest at room temperature at least 30 minutes, and up to 4 hours.
Roll the dough into a log and cut in half. Cut each half into 8 equal pieces (16 total). Roll each into a ball, flatten slightly with your palms, then roll out into a circle 1/8-inch thick.
Place a heavy pot or cast iron pan over medium-high heat and add oil until it is about 1 1/2 inches deep. When the oil shimmers, use tongs to slip in the first poori — it should bubble right away (if not, the oil isn't hot enough). Cook until it puffs and is golden on the underside, then flip and cook golden on the other side (2–3 minutes total). Transfer to a paper-towel-lined plate and repeat with the remaining dough.
Notes
Mind the mustard seeds: They pop and sputter when they hit the hot fat — keep a lid handy.Curry leaves: Add them with the mustard seeds for their distinctive aroma; omit if you can't find them.Rest the poori dough: 30 minutes (up to 4 hours) relaxes the gluten so it rolls thin and puffs.Hot oil for puff: The poori should bubble immediately; if not, the oil needs to be hotter.