Heat 1/2 inch oil in a small heavy skillet over medium heat until shimmering. Fry tortillas one at a time until golden and crisp, 2 minutes per side. Drain on paper towels and sprinkle with salt.
Cool the oil slightly and discard all but 2 tablespoons. Add the beans and cumin and cook, stirring, 1 minute. Add the broth and cook, smashing the beans, until saucy and hot. Season with salt and pepper.
Whisk the garlic, chili paste, 2 tablespoons lime juice, the sugar and 1/2 teaspoon salt in a large bowl. Slowly whisk in 1/4 cup oil to make a dressing; add the cabbage, radishes and cilantro and toss. Mix the sour cream and the remaining 2 tablespoons lime juice in another bowl; season with salt and pepper.
Top each tortilla with beans, chicken, cabbage salad and sour cream.