In a bowl, mash beans into a thick paste and mix in shallot, carrot, quinoa, parsley, feta, salt, and egg whites.
Form mixture into six patties and chill 30 minutes.
Heat 1 tablespoon oil in a skillet over medium-high heat, directly on grill grates or on stove top, and cook patties until golden (add remaining tablespoon oil if cooking in batches), 4 to 5 minutes per side.
Serve on buns with mustard, grilled onions, and radicchio.
For grilled onions, drizzle slices with oil, season with salt and pepper, and grill over medium-high heat, turning, until tender and charred, 6 to 8 minutes.