Pour 1/2 inch oil into an 8- to 10-inch frying pan over medium-high heat. When the edge of a tortilla bubbles vigorously when dipped in the oil, slip tortilla into pan. Cook tortilla, pushing down into oil occasionally and turning over once, until lightly browned, about 1 minute total. Lift out with tongs and transfer to paper towel-lined baking sheets to drain. Repeat to cook remaining tortillas; set in a single layer. As they cool, stack.
In a bowl, mix cabbage, radishes, and lime juice. Add salt and pepper to taste.
Spoon cabbage mixture, refried beans, shredded pork, chipotle salsa, sour cream, cheese, and avocados into separate bowls. Serve with crisp-fried tortillas for guests to assemble their own tostadas: Spread beans on a tortilla, then add pork, cabbage, salsa, sour cream, cheese, and avocado to taste.