Oven & Pan Prep: Position an oven rack in the center and preheat to 425°F (220°C). Place the large rimmed baking sheet in the oven to heat up for at least 10 minutes.
Prepare the Tangy Yogurt Drizzle: While the oven heats, prepare the drizzle. In a small bowl, whisk together ½ cup Greek yogurt, the juice from half the lemon (about 1 ½ tablespoons), and the agave nectar. Season with a pinch of pepper. Set aside in the refrigerator for the flavors to meld.
Season the Vegetables: In a large mixing bowl, combine the asparagus pieces, quartered mushrooms, and whole cherry tomatoes. Add 2 tablespoons of the olive oil, the minced garlic, dried oregano, ½ teaspoon of the sea salt, and ¼ teaspoon of the black pepper. Toss until everything is evenly coated.
Tenderize the Chicken: Place the chicken cutlets between two sheets of parchment paper. Using a meat tenderizer or rolling pin, gently pound them to a uniform thickness of about ½ inch. This ensures they cook evenly.
Create the Savory Coating: In a shallow dish or on a plate, combine the crushed crackers, nutritional yeast, garlic powder, the remaining ½ teaspoon sea salt, and the remaining ¼ teaspoon black pepper. Mix well.
Coat the Chicken: Place ⅓ cup of Greek yogurt in a separate small bowl. Pat the chicken cutlets dry with a paper towel. Dip each cutlet first into the yogurt to coat all sides, then press it firmly into the cracker mixture, ensuring a thick, even crust.
Arrange & Roast: Carefully remove the hot baking sheet from the oven. Arrange the coated chicken on one half of the sheet and spread the seasoned vegetables in a single layer on the other half. Drizzle the remaining 1 tablespoon of olive oil over the chicken. Roast for 22 to 26 minutes, or until the vegetables are tender and the chicken's internal temperature reaches 165°F (74°C).
Final Touches & Serving: Remove the traybake from the oven. Zest the remaining lemon half over the entire pan. Drizzle the chicken and vegetables with the chilled yogurt sauce and garnish generously with chopped fresh parsley before serving.