Preheat the oven to 350ºF. Divide the tortillas between 2 baking sheets and brush them with 1 tablespoon of the olive oil. Bake for 12-15 minutes until golden brown and crisp. Set aside.
Heat the remaining tablespoon of olive oil in a large skillet over medium heat. Add the scallions, corn, jalapeño, and garlic. Cook, stirring occasionally, for 4-5 minutes. Stir in the salsa, cumin, and chicken. Remove from heat.
Divide the mixture evenly among the crisp tortillas. Sprinkle the tostadas with cheese, if using.