Make Pesto. Place basil and garlic in a food processor. Grind until well chopped. Add Parmesan and black pepper. Pulse to create a loose paste. With food processor running, slowly add olive oil until a firm paste forms.
Mix together Wagyu beef, red peppers, and 1/4 cup pesto paste. Form mixture into 4 balls. Flatten each into a patty.
Place 1 ounce of cheese cubes on center of patty. Fold up the sides and pinch shut. Place patties on a medium-high heat grill or griddle pan. Cook for 4 to 5 minutes per side until internal temperature reaches 160 degrees. Serve on a toasted roll with a slice of roasted red pepper and pesto mayo if desired. (make pesto mayo ratio: 1 tsp pesto per 1/4 cup mayo.