Place the potatoes, cauliflower, peas, cabbage, and beetroot in a large pot. Cover with water and bring to a boil. Cook until the vegetables are very tender, about 15 to 20 minutes.
Reserve 2 cups of the cooking liquid, then drain the vegetables. Mash until mostly smooth and set aside.
Heat 2 tablespoons butter and the oil in a large skillet or flat griddle over medium heat.
Add the onion and cook until softened and lightly golden, about 7 to 8 minutes.
Stir in the bell pepper and green chili. Cook for 4 minutes.
Add the ginger-garlic paste and cook for 30 seconds until fragrant.
Add the tomatoes and cook for 10 to 12 minutes, stirring occasionally, until completely softened and jammy.
Stir in 3½ teaspoons pav bhaji masala, the Kashmiri chili powder, turmeric, and salt. Cook for 1 minute.
Add the mashed vegetables and mix well. Mash everything together directly in the skillet for a smooth texture.
Gradually add the reserved cooking liquid until the mixture reaches a loose, spoonable consistency.
Simmer for 20 to 25 minutes, stirring frequently and mashing occasionally.
During the last 5 minutes of cooking, stir in 1 tablespoon of the remaining butter.
Remove from the heat and stir in the remaining 2 tablespoons butter, the remaining ½ teaspoon pav bhaji masala, lemon juice, and cilantro.
Taste and adjust seasoning if needed.
Slice the pav buns horizontally.
Melt 2 tablespoons butter on a griddle or skillet over medium heat. Toast the pav buns until golden brown on both sides.
Serve the hot bhaji topped with chopped onion, cilantro, a small pat of butter, and lemon wedges alongside the toasted pav.