Preheat the oven to 375°. In a medium saucepan, heat 2 tablespoons oil over medium heat. Add the garlic and cook, stirring, until just golden, about 2 minutes. Stir in the beans and cook for 1 minute, then stir in the chicken broth. Season with salt and cook, mashing the beans, until softened, 7 minutes. Let cool slightly.
Meanwhile, line a baking sheet with paper towels. In a small skillet, heat the remaining 1/2 cup oil. Fry each corn tortilla for 3 seconds on each side, then transfer to the prepared baking sheet to drain.
Place about 1/4 cup mashed beans on 1 tortilla and roll up to enclose; transfer to an oiled 9-by-13-inch baking dish. Repeat with the remaining beans and tortillas.
Pour the salsa over the filled tortillas. Sprinkle the cheese on top. Bake until the cheese is melted and the salsa is bubbling, about 20 minutes.