Trim the chicken breasts of any excess fat and pat them dry with paper towels—this helps them brown better on the grill. In a small bowl, combine the garlic powder, Mediterranean oregano, kosher salt, and black pepper. Generously coat both sides of the chicken breasts with this seasoning mixture, pressing gently so it adheres. Set the seasoned chicken aside at room temperature while you prepare the vegetables. I like to season the chicken at least 10 minutes before grilling so the flavors start to penetrate the meat.
While the chicken seasons, trim the green beans and snap them into 2-inch pieces. Halve the cherry tomatoes lengthwise and smash the garlic cloves with the side of your knife. In a large bowl, combine the prepared green beans, tomatoes, and smashed garlic. Drizzle with the Bertolli Extra Virgin olive oil and toss to coat evenly. Season with a pinch of kosher salt and freshly ground black pepper, then set aside. This vegetable mixture is ready to grill whenever you need it.
Line half of your grill grates with aluminum foil to create an indirect heat zone—this will be perfect for the vegetables so they don't char too quickly. Heat the grill to medium-high (about 400°F). Once the grill reaches temperature, oil the bare grill grates where you'll place the chicken using a folded paper towel dipped in oil or a grill brush. The foil-covered side will hold the vegetables and won't need oiling.
Place the seasoned chicken breasts on the oiled grill grates and cook for 7-8 minutes without moving them—this creates a nice golden crust. While the chicken cooks, spread the green bean and tomato mixture from Step 2 on the foil-lined side of the grill. Flip the chicken and cook for another 7-8 minutes until the internal temperature reaches 165°F. The vegetables should finish around the same time, becoming tender with light charring on the tomatoes. I find that the vegetables and chicken finish at almost exactly the same time if you start them together, which makes plating much easier.
Transfer the grilled vegetables from Step 4 to a clean bowl. While they're still warm, drizzle with red wine vinegar and toss gently to combine. Add the chopped fresh dill and red pepper flakes, stirring to distribute the seasonings evenly. Taste and adjust seasoning with salt and pepper if needed. The warm vegetables will absorb the vinegar beautifully, creating a bright, Mediterranean flavor.
Divide the grilled chicken breasts among serving plates. Arrange the dressed vegetables from Step 5 alongside the chicken. Crumble the feta cheese over the top of each plate, then drizzle everything with extra-virgin olive oil. Finish with a light sprinkle of fresh lemon zest for brightness. Serve immediately while everything is still warm.