Siblings don't always agree, but my kids agree wholeheartedly that this is the best sandwich! The classic combination of mozzarella, tomato, and basil is delicious, and the balsamic glaze adds a sweet tang.
To make the glaze: Combine the vinegar and sugar in a small saucepan and bring to a boil. Turn the heat down to medium and let simmer for 4 to 6 minutes, stirring constantly with a wooden spoon, until the volume has been reduced by half. Set aside to cool in a heat-proof container.
To make the pesto: Place the basil, pine nuts, Parmesan, garlic, lemon juice, and salt in a food processor. Pulse for about 1 minute, until the ingredients look chopped. Add the olive oil and blitz until smooth.
To make the sandwiches: Use the butter to grease a skillet.
Cut a roll in half horizontally and slather 2 tablespoons of pesto onto each side. Layer mozzarella over the pesto on the bottom half of the roll. Layer tomatoes over the cheese. Drizzle 1/2 teaspoon of glaze over the cheese. Cover with basil leaves, and close the sandwich with the top of the roll. Repeat with each roll.
Warm the skillet over medium heat. Place the sandwiches in the skillet and cover with a lid. The goal is to lightly brown both sides of the sandwich and melt the cheese. This takes about 3 minutes of cooking time on each side. Remove the sandwiches from the skillet, cut in half, and serve warm.