To make the green marinade/sauce: Add 1 cup cilantro, 1 cup parsley, 4 scallions, 2 garlic cloves, 1/3 cup olive oil, juice of 1 lime, juice of 1 lemon, and 1/2 teaspoon salt to a blender and blend until smooth.
Place 1 pound chicken breasts, 1 pound chicken thighs, 1 thinly sliced zucchini, 1 thinly sliced yellow squash, and 1 quartered red bell pepper into a large flat baking dish or gallon Ziploc bag and cover with half of the green sauce. Reserve the second half of the sauce for serving. Yes - the vegetables and chicken can touch even though the chicken is raw! We are cooking them! It’s all good!
Toss the chicken and vegetables to coat. Cover and refrigerate for up to 24 hours, flipping occasionally, or marinate at room temperature for 15 minutes, flipping once.
Preheat the grill to medium-high heat.
Shake off as much of the marinade from the chicken and vegetables as possible as you add them to the grill, to prevent flare-ups. Grill the chicken thighs and peppers for 5 minutes per side, breasts for 4 minutes per side, and squash and zucchini for 3 to 4 minutes per side, or until charred and tender. Transfer everything to a large platter as it finishes.
Garnish the platter with 4 ounces burrata, torn into 2 or 3 pieces and 2 cups arugula, and drizzle it all with the reserved green sauce. Serve the remaining green sauce in a bowl on the side. Enjoy hot or at room temperature.