Make the vinaigrette by combining 2/3 cup basil pesto, 1/3 cup balsamic vinegar, 3 tablespoons honey, 2 tablespoons lemon juice, and 2 tablespoons extra virgin olive oil in a jar or bowl; season with chili flakes, kosher salt, and black pepper to taste and whisk until smooth.
Place the chicken chunks in a bowl and toss with half of the vinaigrette and 1 tablespoon extra virgin olive oil; let marinate briefly while you prepare the tomatoes.
In a separate bowl, toss the cherry tomatoes with 1 tablespoon extra virgin olive oil and a pinch of salt and pepper.
Thread the marinated chicken and tomatoes onto skewers, alternating pieces; if you have extra tomatoes or chicken make additional skewers as needed.
Preheat a grill, grill pan, or skillet to medium-high. Grill the skewers, turning occasionally, until the chicken is lightly charred and cooked through, about 10 to 12 minutes total.
Toss the mozzarella balls with a couple tablespoons of the reserved vinaigrette. Arrange the mozzarella and room-temperature burrata on a serving plate and place the hot skewers on top.
Spoon the remaining vinaigrette over the skewers and cheese, drizzle additional olive oil or balsamic if desired, and scatter fresh basil leaves over everything before serving.