In a small bowl mix together the chipotle chili powder and mayonnaise until thoroughly combined. Set aside.
Slice the chorizo into thin slices.
Line a heat-proof plate with paper towels. Heat a large skillet over medium heat, and then add 1 tablespoon of the oil. Swirl the oil in the pan to coat it. Add in the sliced chorizo. Cook over medium heat for about 10 minutes, stirring often until the meat is browned. Transfer the chorizo to a paper-towel lined plate, and wipe the pan clean with a paper towel.
Lay three of the tortillas out on your work surface. Spread each with about 1 tablespoon of the chipotle mayo. Divide the cooked chorizo into three portions and spread a portion over each tortilla.
Sprinkle 3/4 of a cup of the shredded cheese over the top of one of the tortillas with chorizo. Repeat with the remaining two tortillas.
Lay out the last three tortillas and spread each with tablespoon of the chipotle mayonnaise. Place them mayo-side down on top of each of the assembled quesadillas.
Preheat the oven to 200°F. Place a baking sheet inside.
Heat the skillet over medium heat, and add 1 tablespoon of the oil. Swirl the oil in the pan to coat it. Place one of the assembled quesadillas in the pan. Press down the top with a spatula and cook for two minutes. Carefully flip the quesadilla over and cook for two more minutes - both sides should be lightly browned.
Remove the quesadilla with a spatula and place it on the baking sheet in the oven to stay warm while you cook the last two.
Repeat the steps to cook the remaining two quesadillas.
Slice each quesadilla into fourths and serve immediately.