Place the potatoes, carrot, green beans, and peas in a pot with enough water to cover. Bring to a boil, reduce heat to low, cover, and simmer 15 minutes, until potatoes are tender.
Drain vegetables, transfer to a bowl, and mash with a potato masher, leaving a few small chunks.
Mix the egg, 3/4 cup bread crumbs, garam masala, cayenne pepper, and salt into the bowl with the mashed vegetables. Form the mixture into small patties about 1/2 inch thick. Dredge the patties in the remaining bread crumbs to coat.
Heat the oil in a large skillet over medium heat, and fry the coated patties until golden brown.
