Prepare a grill for medium heat.
Meanwhile, toss the chicken with half the lemon juice, 2 tablespoons of the olive oil, the oregano, 1 teaspoon salt and a few grinds of black pepper. Let it sit to marinate for about 10 minutes (don’t let it sit any longer or the acid from the lemon juice will start to cook the chicken).
While the chicken is marinating, toss the tomatoes, shallots, garlic, 2 tablespoons of the olive oil, 1 teaspoon salt and a few grinds of black pepper in a medium bowl. Pour the mixture into the middle of two large sheets of foil stacked on top of each other. Place one ball of burrata on top of the tomatoes, fold the foil over all the ingredients and pinch it tightly on all sides to seal it closed. Grill until the tomatoes are starting to burst and the burrata is melted and gooey, 14 to 16 minutes.
Place the chicken on the grill and cook until dark grill marks appear on the bottom, 7 to 8 minutes. Flip the chicken and continue cooking until a digital thermometer inserted into the thickest part of the chicken registers 165 degrees, 7 to 8 minutes. Remove from the grill and let it rest for 5 minutes. Slice the chicken into 1/2-inch-thick pieces.
While the burrata and chicken are grilling, bring a large pot of salted water to boil. Cook the cavatappi until al dente, about 7 minutes. Reserve 1/4 cup of the pasta cooking water, then drain and set aside.
Meanwhile, add the panko, 2 tablespoons of the olive oil, 1 teaspoon salt and a few grinds of black pepper to a large skillet and cook over medium-high heat, stirring occasionally, until the panko is golden brown, 5 to 6 minutes. Stir in the lemon zest and set aside. Toss the arugula with the remaining lemon juice and remaining 1 tablespoon olive oil.
Empty the contents of the foil packet into the drained pasta and stir until well combined. Fold in the Parmesan, chicken and reserved pasta water until the cheese has melted. Transfer the pasta to a serving bowl and top with the breadcrumbs. Place the arugula on top of the breadcrumbs and then top with the remaining ball of burrata. Drizzle with balsamic and slice into the burrata so that the soft center oozes out. |