Rinse, peel, and chop the vegetables. Pressure cook the potatoes, cauliflower, carrot, peas, and beans with 2.25 to 2.5 cups water for 5 to 6 whistles, about 12 minutes on medium heat (Veg Recipes of India). For an Instant Pot, pressure cook 6 minutes on high, then quick release (Ministry of Curry). Reserve the cooking stock.
Heat a wide pan or kadai and melt 3 tablespoons butter. Add cumin seeds, then the onion. Sauté until translucent, about 5 minutes. Stir in ginger-garlic paste and green chili, then the capsicum, and sauté 3 to 4 minutes (Indian Healthy Recipes).
Add the chopped tomatoes and sauté on low to medium heat until soft and mushy and the butter separates at the sides, about 6 to 7 minutes (Veg Recipes of India).
Add turmeric, Kashmiri chili powder, and pav bhaji masala. Mix well and sauté about 2 minutes, until the raw smell goes (Indian Healthy Recipes). For deeper red without heat, stir in the soaked-chili paste here.
Add the cooked vegetables and their stock. Season with salt. Mash directly in the pan with a potato masher for a coarse texture, or use an immersion blender for a smooth paste. Add water if it looks thick; the consistency should be neither thin nor stiff.
Simmer on low, stirring occasionally, for 8 to 10 minutes (Veg Recipes of India) - up to about 15 minutes for a deeper flavor (Cook With Manali). Taste; for more heat add another ½ teaspoon chili powder and cook 3 to 4 minutes more (Indian Healthy Recipes).
Slice each pav horizontally. Heat a griddle or skillet on low, add butter, sprinkle a little pav bhaji masala into it, and press the cut sides of the rolls down until warm and lightly toasted, so they absorb the spiced butter.
Serve the bhaji hot, topped with a cube of butter, chopped raw onion, cilantro, and a lemon or lime wedge, with the toasted pav alongside.