Combine the arugula, walnuts, cheese, and garlic in the bowl of a food processor. Pulse until the mixture is finely chopped, then pour olive oil through the feed tube and continue to process until smooth. Season to taste with salt and pepper.

Preheat oven to 500ºF or broil.

Line a sheet pan with foil. Place the peppers on the pan and broil them until the skin begins to char, turning occasionally, about 10 minutes. Remove the pan from the oven and using oven mitts, wrap the foil around the peppers. Steam the peppers in the foil for 10 minutes.

Carefully remove the peppers from the foil and transfer them to a cutting board to cool slightly. Once they're cool enough to work with, peel off the skin, cut off the stem, scrape out the seeds, and discard them. Cut the rest of the peppers into thick strips.

Spread the pesto onto the bottom halves of the bread. Divide the pepper strips over the pesto, then place the mozzarella on top of the pepper. Top with the remaining slices of bread and brush the tops with olive oil.

Heat a grill pan or griddle over medium heat. Flip the sandwiches over and place them on the grill pan oiled-side down, then brush the other side of the sandwich with oil. Use a spatula to press the sandwiches down slightly. Cook 5-8 minutes on each side, or until the bread is toasted and golden brown and the cheese is melted.