Preheat the grill to medium-high heat (about 400°F).
In a small bowl, combine 1 tablespoon olive oil, garlic powder, smoked paprika, salt, and black pepper.
Brush the chicken breasts evenly with the seasoned olive oil mixture on both sides.
In a large bowl, toss zucchini, red bell pepper, yellow bell pepper, and cherry tomatoes with the remaining 1 tablespoon olive oil and balsamic vinegar. Season with a pinch of salt and pepper.
Place the chicken breasts on the grill and cook for 6-8 minutes per side, or until the internal temperature reaches 165°F and the chicken is cooked through.
At the same time, place the vegetables in a grill basket or on a grill pan and cook for about 10-12 minutes, turning occasionally, until tender and slightly charred.
While the chicken and vegetables are grilling, prepare the salad by combining mixed salad greens, cucumber, shredded carrots, and sliced red onion in a large bowl.
In a small bowl, whisk together lemon juice, Dijon mustard, salt, and black pepper to make the dressing.
Drizzle the dressing over the salad and toss to combine.
Remove the chicken and vegetables from the grill and let the chicken rest for 5 minutes.
Serve the grilled chicken breasts with the roasted vegetables and fresh garden salad on the side.