Pulse beans in a food processor until mixture resembles ground meat. Pulse in rice. Stir bean mixture in a bowl with carrot, broccoli, ginger, cilantro, and flavor base. Form into twelve 2 1/2-inch round patties (1/4 cup each). Cook in hot oil over medium-high heat until golden, about 3 minutes per side. Sandwich in rolls, adding garnishes.
For the Chili Mayonnaise: Stir together 3/4 cup mayonnaise, 1/2 teaspoon Asian chili paste (such as sambal oelek), and a dash of toasted-sesame oil.