In a small bowl, whisk together the mayonnaise, sour cream, minced pickled jalapeños, jalapeño brine, cumin, garlic powder, paprika, salt, and cayenne until smooth and fully combined. Taste and adjust seasoning, then cover and refrigerate while you prepare the chicken.
If the chicken breasts are thicker than 3/4 inch (2 cm), pound them to an even 1/2-inch (1.25 cm) thickness. Drizzle olive oil over both sides, then rub the combined chili powder, smoked paprika, cumin, garlic powder, onion powder, cayenne, salt, and pepper onto all sides.
Heat the skillet over medium-high heat for 2 minutes, add a drizzle of oil, then cook the chicken for 5–6 minutes per side until the internal temperature reaches 165°F (74°C). Transfer to a cutting board, rest for 5 minutes, then slice into thin strips and roughly chop into bite-sized pieces.
Lay one tortilla flat, spread 1–1.5 tablespoons of jalapeño sauce over one half, then top with one-quarter of the chopped chicken and 1/2 cup (55g) of shredded cheese. Fold the empty half over the filled half to form a half-moon and press lightly to seal.
Melt 1/2 teaspoon of butter in the wiped-out skillet over medium heat, add one assembled quesadilla, and cook for 2–3 minutes per side until deep golden brown and the cheese is fully melted. Repeat with the remaining quesadillas, adding butter before each one.
Rest each quesadilla on a cutting board for 60 seconds, then slice into 3 equal wedges using a sharp knife or pizza cutter. Serve immediately with extra creamy jalapeño sauce on the side for dipping.