Cut tortillas in half with a pizza cutter. This step is optional, but makes the tostadas easier to pick up by hand.
Cover the bottom of a large frying pan with oil that has a high smoke point, such as canola, peanut or avocado oil, to a depth of about 1/2 inch.
Heat the oil over medium high heat, until it shimmers slightly. If you dip the edge of a tortilla into the oil and it begins to bubble and sizzle, the oil is ready for frying. If you are using a frying thermometer, heat the oil to 375 degrees.
Using tongs, carefully place 3-4 tortilla halves into the oil. Fry for about one minute on each side until browned and crispy.
Remove tostada shells from oil and drain on paper towels. Sprinkle lightly with salt. Repeat until all the tortillas are fried.
If the tostada shells are chewy and not completely crisp, finish by baking in a 350 degree oven for 5-8 minutes.
Prepare the filling by cooking the chorizo and onion in a frying pan over medium high heat until the chorizo is browned and cooked through. Add the refried beans, stir to combine, and continue cooking until hot.
Top each half tostada shell with about 2 tablespoons of bean and chorizo filling.
Add cheese and other toppings as desired. If you wish, you can let each person top their own tostadas at the table.