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Start Traeger on Smoke with the lid open until fire is established (4 to 5 minutes).
Preheat grill to 350 degrees F with the lid closed for 10 to 15 minutes.
Rinse and pat dry chicken.
Create a deep pocket in each chicken breast with sharp knife.
Combine quartered tomatoes, mozzarella, garlic,1 tbsp olive oil, basil, salt, pepper and lemon juice and gently toss together.
Stuff each chicken breast equally with the tomato mixture.
Transfer chicken to a foil-lined baking sheet (sprayed with nonstick cooking spray).
Drizzle chicken lightly with olive oil and balsamic glaze and season generously with salt and pepper.
Distribute 1 cup of whole cherry tomatoes around chicken.
Place the baking sheet on the grill grate and cook for 25 minutes, or until chicken is completely cooked (internal temperature should read 165 degrees on an instant-read thermometer).
Garnish chicken with more fresh basil and serve. Enjoy!