Butter one side of each piece of bread with about 1/2 tbsp of butter each. Lay out bread buttered-side down. Layer with cheese, pesto, cheese and top with bread, buttered-side up. Continue until all sandwiches are made.
Heat up a griddle or a skillet over medium heat. Place sandwiches onto heated surface (as many as can fit) and press down throughout the grilling process with a spatula. Flip once the first side is golden, ~ 3 to 4 minutes.
Whisk together maple syrup, balsamic vinegar, and oil. Season to taste with salt and pepper. Dress arugula in vinaigrette and garnish with feta (if using).
Enjoy grilled cheese sandwiches with salad on the side.