Make Pesto
Fit food processor with steel blade attachment. Start motor, and drop garlic through the feed tube. Process until the garlic is chopped (it’s okay if it stick to the sides.) Remove lid. Add 2 cups spinach, nuts, 3 tablespoons extravirgin olive oil, Parmesan-style cheese, 2 tablespoons water and salt. Process until pureed, about 10 seconds. Scrape sides and process until smooth, about 20 seconds.

Make Quesadillas
Combine 4 cups spinach and 2 tablespoons water in a large microwave-safe bowl or measuring cup. Cover and microwave until the spinach is wilted, about 3 minutes on high. Carefully remove cover and drain the spinach.

Lay tortillas on work surface. Spread pesto over half of each tortilla, dividing evenly among the 4. Top the pesto with spinach, dividing among the tortillas. Top the spinach with cheddar. Fold tortillas in half.

Brush 1 teaspoon oil over a griddle. Set quesadillas on the griddle and set over medium heat. Cook until the bottoms are browned and crisped, 4 to 7 minutes. Flip the quesadillas and continue cooking until the other side is browned and the cheddar cheese is melted.

Alternatively, to cook in a large skillet
Oil a large skillet, add 2 quesadillas and set over medium-high heat. Cook until the bottom is browned, 6 to 7 minutes. Reduce heat to medium and continue cooking until the other side is browned, 2 to 3 minutes. Reduce temperature to medium-low and repeat with the remaining 2 quesadillas, cooking them until brown, about 2 minutes per side. Cut into wedges to serve.