If using wooden skewers, soak them in water for at least 1 hour before grilling.
Preheat a grill or grill pan to medium-high (about 425°F / 220°C).
Cut the chicken breasts into pieces suitable for skewering so each skewer will hold about 4 pieces.
Thread the chicken pieces and grape tomatoes onto skewers, aiming for about 4 pieces of chicken and 3 tomatoes per skewer.
Brush both sides of the skewers with 1 tablespoon olive oil and season with salt and freshly ground black pepper.
Grill the kebabs 4–6 minutes per side, flipping once, until the chicken reaches 165°F (74°C) in the center.
While the kebabs cook, make the basil pesto: in a food processor pulse the basil, Parmesan, pine nuts (or walnuts), and minced garlic until finely chopped.
With the processor running, slowly pour in 1/3 cup olive oil until the mixture is combined; season to taste with salt and pepper.
When the kebabs are cooked through, remove them from the grill and thread one mozzarella ball onto each end of the skewer.
Spoon the basil pesto over the kebabs and drizzle with 3 tablespoons balsamic glaze; serve warm.