Chop potatoes, tomatoes, and mixed vegetables into small uniform pieces and measure out spices and other ingredients.

Heat oil in a large saucepan over medium heat, add cumin seeds, and cook until fragrant and slightly darkened, about 1 minute.

Add minced garlic and grated ginger to the pan; sauté until fragrant, about 30 seconds, with a sizzling sound.

Stir in chopped tomatoes, cook until they soften and mixture thickens, about 5-7 minutes, watching for bubbling and reduced volume.

Add spices: turmeric, chili powder, garam masala, and salt; cook for 1 minute, ensuring spices release aroma and blend well.

Add chopped potatoes and mixed vegetables; stir well, then pour in enough water to cover vegetables, simmer uncovered until vegetables are tender and mixture thickens, about 15-20 minutes, watching for bubbling and thickened consistency.

While bhaji simmers, slice buns in half and toast in butter on a skillet over medium heat until golden brown and crispy, about 2-3 minutes, with sizzling sound and golden color.

Serve hot bhaji topped with chopped cilantro and a squeeze of lemon, with toasted buns on the side, finishing with a pat of butter on each bun for extra richness.