Place the rinsed tomatillos and jalapeño on a rimmed baking sheet. Broil on high (or roast at 450°F/230°C) for 5–8 minutes, turning once, until the tomatillos are softened and lightly charred and the jalapeño skin is blistered.

Blend salsa

Transfer the hot tomatillos and jalapeño to a blender or food processor. Add the chopped onion, cilantro, lime juice, 1/2 tsp salt, and 2 tbsp water. Pulse/blend until mostly smooth. Taste and adjust seasoning with more salt or lime juice as desired. If salsa is too thick, add additional water 1 tsp at a time to reach a spoonable but not watery consistency. Let cool to room temperature while preparing quesadillas.

Quesadilla

Prepare tortillas and cheese

Shred the cheddar-jack cheese if not pre-shredded. Lay out 4 tortillas. Divide the shredded cheese evenly among two tortillas (about 1 cup cheese each), leaving a 1/4-inch border.

Assemble and cook

Heat a large nonstick skillet over medium heat. Add 1 tsp olive oil and 1/2 tbsp butter and swirl to coat. Place one tortilla (cheese side up) in the skillet and top with one of the cheese-covered tortillas (cheese sides facing each other) to create a stacked quesadilla, or place cheese on single tortilla and fold another over. Cook 2–3 minutes until the bottom tortilla is golden brown and cheese begins to melt. Carefully flip with a spatula and add remaining 1/2 tbsp butter to the pan; cook another 2–3 minutes until golden and cheese is fully melted. Repeat with the remaining tortillas, adding butter as needed.

Finish and serve

Transfer quesadillas to a cutting board and let rest 1 minute. Cut each quesadilla into 4 wedges with a sharp knife or pizza cutter. Season wedges lightly with 1/4 tsp salt and 1/8 tsp black pepper distributed across both quesadillas if desired. Serve warm with salsa verde on the side for dipping or spooning over the wedges.