Use a spiralizer to turn the zucchini into noodles. Place the noodles in a colander, sprinkle lightly with salt, and let them sit for 10 minutes to release excess moisture. Pat dry with a clean towel.

Season the chicken breasts with salt and black pepper on both sides. Heat 1 tablespoon olive oil in a grill pan over medium-high heat. Grill the chicken for 6 to 7 minutes per side until cooked through. Remove from heat and let rest for 5 minutes before slicing.

While the chicken rests, heat the remaining 2 tablespoons olive oil in a large skillet over medium heat. Add the minced garlic and cook for 30 seconds until fragrant.

Add the sliced bell pepper and asparagus pieces to the skillet. Cook for 4 to 5 minutes, stirring occasionally, until the vegetables begin to soften.

Add the zucchini noodles and cherry tomatoes to the skillet. Cook for 2 to 3 minutes, tossing gently, just until the noodles are warmed through but still firm.

Remove the skillet from heat. Stir in the chopped basil, chopped parsley, and grated Parmesan cheese. Toss to combine evenly.

Divide the zucchini noodle mixture among four plates. Top each serving with sliced grilled chicken.