If using corn tortillas instead of store-bought shells, preheat the oven to 400°F (205°C). Brush tortillas lightly with oil on both sides, arrange on a baking sheet in a single layer, and bake 5–7 minutes per side until crisp and golden. Cool slightly.
In a large skillet over medium heat, warm the olive oil. Add the shredded chicken and BBQ sauce and cook, stirring, until hot throughout (about 3–5 minutes). Taste and add a little more sauce if you want it saucier.
Set out tostada shells and toppings. For each tostada, spoon on BBQ chicken, then layer with lettuce, tomatoes, black beans, corn, red onion, and cheese.
Add optional toppings as desired (sour cream/Greek yogurt, avocado or guacamole, cilantro, lime, jalapeños, pico de gallo, or a thin layer of refried beans). Serve immediately so the shells stay crunchy.