In a frying pan, warm oil over medium heat. Add garlic, onion, and red pepper and cook, stirring occasionally, until onion is softened, 8 to 10 minutes. Stir in ground chile and cumin and cook until fragrant, about 1 minute.
Drain beans, reserving their liquid, and add to pan along with about 1/2 cup liquid. Stir and mash until beans form a slightly chunky purée and are heated through, about 3 minutes.
To assemble tostadas, top each shell with 1/2 cup bean mixture, a few radishes, avocado slices, and cheese. Serve with lime wedges and salsa and/or sour cream.
*Find in Latino markets and well-stocked grocery stores.