Cook scallions, carrot, celery, garlic, thyme, salt, and pepper in 2 tablespoons oil in a 12-inch heavy nonstick skillet over moderate heat, stirring, until carrot is softened, about 8 minutes. Stir together rice, eggs, bread crumbs, and vegetable mixture in a bowl. Form into 8 (2 1/2- by 1/2-inch) patties, pressing mixture so cakes will hold together (mixture will be loose). Transfer to wax paper as formed. Cook 4 rice cakes in 2 tablespoons oil in skillet over moderately high heat, gently turning over once, until browned, about 7 minutes total. Transfer to a plate and keep warm, loosely covered with foil. Cook remaining 4 cakes in remaining tablespoon oil.