Boil the potatoes and cauliflower in lightly salted water for 12 to 15 minutes until tender. Add the peas for the final 3 minutes, then drain.
Heat the oil and 2 tablespoons butter on a wide skillet. Cook the yellow onions and bell pepper for 8 minutes. Add garlic and ginger for 1 minute, then tomatoes, pav bhaji masala, chile powder, turmeric, and salt.
Mash the Bhaji:
Cook the tomato mixture for 10 minutes until soft. Add the drained vegetables and water. Mash thoroughly while simmering until cohesive but not perfectly smooth.
Cook for 10 to 15 minutes, stirring and adding a little water as needed. Beat in 2 tablespoons butter and taste for seasoning.
Toast and Serve:
Spread the remaining butter on the cut rolls and toast them cut-side down until unevenly golden.
Serve the bhaji hot with pav, red onion, cilantro, and lime wedges.
Notes
Mash while the vegetables are hot for the smoothest texture.
Pav bhaji masala varies by brand; add extra chile only after tasting.