Wash and dry the basil leaves thoroughly. Toast the pine nuts in a dry skillet over medium heat for 2–3 minutes until fragrant and lightly golden, stirring constantly. In a food processor or high-speed blender combine basil, toasted pine nuts, grated Parmesan, garlic clove, lemon juice, salt, and pepper. Pulse to combine.

With the processor running, slowly drizzle in the extra-virgin olive oil until the mixture emulsifies into a smooth, spoonable pesto. Taste and adjust salt or lemon as needed. Transfer pesto to a small bowl and set aside. (Yield about 1/2 cup)

Quesadilla Assembly & Cooking

Prepare tomatoes

Core and thinly slice the tomatoes into 1/8–1/4-inch slices. Lay slices on paper towel and lightly season with 1/2 teaspoon salt and 1/4 teaspoon black pepper; let drain for 5 minutes to release excess moisture, then pat dry.

Assemble quesadillas

Spread about 1/2 tablespoon of the basil pesto evenly over one side of each tortilla (reserve remaining pesto for serving or additional spread). On half of each tortilla, sprinkle about 1/2 cup shredded cheddar-jack cheese, arrange tomato slices in a single layer on top of the cheese, then add another light sprinkle (~2 tablespoons) of cheese to help bind. Fold the exposed half of the tortilla over the filling to make a half-moon.

Cook

Heat a large nonstick or cast-iron skillet over medium-low to medium heat. Add 1/2 tablespoon butter and swirl until melted and foaming. Place one assembled quesadilla in the skillet, cook 2–3 minutes until the bottom is golden brown and crisp. Carefully flip (use a spatula) and add another 1/2 tablespoon butter around the edge of the quesadilla; cook an additional 2–3 minutes until the second side is golden and the cheese is fully melted. Reduce heat slightly if tortilla browns too quickly. Repeat with remaining quesadillas, adding butter as needed (total about 2 tablespoons).

Finish and serve

Remove each quesadilla to a cutting board and let rest 1 minute. Slice into 3 wedges with a sharp knife or pizza cutter. Garnish with a small dollop of extra pesto if desired and a light sprinkle of red pepper flakes for heat. Serve immediately while hot and melty.