If chicken breasts are not thin-sliced, carefully slice in half lengthwise. In a medium bowl, coat chicken with 1 Tbsp. oil. Coat all sides with chili powder and cumin and season with salt and pepper. Let sit while you prepare the vegetables.

Heat a Carbon Steel Griddle over medium-high. Add bell pepper, onion, and jalapeño to a medium bowl, toss with 1 Tbsp. oil and season with salt and pepper. Drizzle 1 Tbsp. oil on Griddle, add vegetables, and cook, tossing occasionally, until softened and blackened, 4–6 minutes. Add corn and garlic and continue to cook, tossing, until corn is charred in spots and garlic is fragrant, 1–2 minutes more. Transfer vegetables to a large Platter or bowl to cool slightly.

Heat Griddle over medium and drizzle with 1 Tbsp. oil. Add chicken and cook until deeply browned and cooked through to 165F internal temperature, 4–5 minutes per side. If you have a Griddle Lid, cover chicken to help cook quickly and evenly. Transfer to a Butcher Block and let cool slightly. Cut into ½” cubes and add to bowl with vegetables. Chicken and vegetable mixture can be made up to 3 days in advance; gently warm before assembling quesadillas.

Scrape Griddle to remove any residue. Return heat to medium. Drizzle with about 1 Tbsp. oil and add 6 tortillas, topping each with about ¼ cup cheese, spreading cheese all the way to the edges. Place about ¼ cup chicken mixture on one half of each tortilla. Cook until cheese is melty and tortilla is golden brown and crispy on the bottom, 3–5 minutes, then carefully fold tortilla over, pressing gently to create half moon shape. Transfer to a Platter to cool slightly.

Repeat process with remaining tortillas, cheese, and chicken mixture, adding additional oil as needed.

Top quesadillas with pickled red onion and cilantro, if desired, and serve.