Preheat your Blackstone griddle to medium heat (for cooking the veggies) and medium-low heat on the other side (for toasting the quesadillas).
Drizzle olive oil onto the medium-heat zone. Add the diced bell pepper and red onion. Sauté for 3-4 minutes until they are softened.
Add the pre-cooked shredded chicken to the vegetables, sprinkle with taco seasoning, and toss everything together until warmed through. Move this mixture to a cooler side of the griddle.
On the medium-low heat zone, melt half of the butter and spread it around. Place two flour tortillas onto the buttery surface.
Quickly sprinkle a generous handful of grated cheese over the entire surface of each tortilla. Spoon the warm chicken and vegetable mixture onto only one half of each tortilla.
As soon as the cheese begins to melt (about 1-2 minutes), use your spatula to fold the empty, cheesy half of the tortilla over onto the filled half, creating a half-moon shape.
Press down gently with your spatula. Cook for another 1-2 minutes until the bottom is perfectly crispy and golden brown.
Carefully flip the folded quesadillas and cook the other side for 1-2 minutes until both sides are golden and the cheese is completely melted and gooey inside. Repeat the process with the remaining tortillas and butter.
Remove from the griddle, let them rest for a minute so the cheese sets slightly, then cut into triangles. Serve hot with sour cream and salsa!