Dice 1/2 onion, 2 zucchini, and chop 4 cloves garlic. Cut 4 cups broccoli into bite-sized florets. Pound 2 boneless skinless chicken breasts to even thickness and season both sides with salt, garlic powder, and pepper. Measure 3-1/2 cups chicken broth and pour 1 cup into a separate bowl, leaving 2-1/2 cups in a high-speed blender pitcher.
Heat 2 tablespoons light olive oil in a large Dutch oven or stock pot over high heat. Add diced zucchini and onion, stirring frequently for 5-7 minutes until edges begin to turn golden and caramelize. Add chopped garlic during the last minute to bloom and become fragrant. Remove vegetables from pan and add to the blender with the 2-1/2 cups chicken broth.
Add 2 more tablespoons light olive oil to the same pot. Lay in seasoned chicken breasts and cook for 3 minutes on each side until golden brown. Slowly pour in the reserved 1 cup chicken broth, scraping up any browned bits from the bottom of the pan for extra flavor.
Blend the caramelized vegetables with broth until completely smooth. Pour this mixture back into the pot with the chicken. Add 1 tablespoon dried thyme and 1 teaspoon salt, stirring to combine. Cover, reduce heat to medium, and simmer for 10 minutes.
Add broccoli florets to the pot, submerging them in the liquid. Cover and simmer for 8 minutes or until broccoli is fork-tender but still has some bite. Remove from heat and transfer chicken to a cutting board.
Stir 1 cup Greek yogurt into the hot soup until completely melted and dissolved, about 1 minute. Shred the chicken with two forks and return to the soup, stirring well. Serve immediately topped with shredded white cheddar cheese and minced fresh parsley.