Marinate the chicken. In a bowl, mix 1 tablespoon olive oil, half the lemon zest, half the lemon juice, a pinch of salt, and black pepper.
Add the chicken and toss to coat. Let it sit for 15–20 minutes at room temperature while you prep the salad.
Make the dressing. In a small jar or bowl, whisk together 2 tablespoons olive oil, remaining lemon zest and juice, balsamic vinegar, honey, and garlic. Season with 1/2 teaspoon salt and a few grinds of pepper.
Taste and adjust—add a splash more balsamic for tang or a pinch more honey for balance.
Cook the chicken. Heat a grill pan or outdoor grill over medium-high. Cook the chicken 5–7 minutes per side, depending on thickness, until the center is no longer pink and juices run clear (internal temp 165°F/74°C). Let rest on a plate for 5 minutes, then slice.
Prep the produce. While the chicken cooks, halve the tomatoes, slice the peaches and cucumber, and shave the red onion.
Tear the basil just before using so it stays fragrant.
Assemble the greens. Add arugula and spinach to a large bowl or platter. Drizzle with about half the dressing and toss lightly so the leaves are just coated. Don’t drench them—you’ll add more dressing at the end.
Add the colorful bits. Scatter tomatoes, peach slices, cucumber, onion, and most of the basil over the greens. If using nuts, toast them in a dry skillet for 2–3 minutes until fragrant, then sprinkle on top.
Finish with chicken and burrata. Arrange the sliced chicken over the salad.
Gently tear each burrata ball open and nestle the creamy centers onto the greens. Season the burrata with a pinch of salt and pepper.
Dress and serve. Spoon the remaining dressing over the chicken and burrata. Add a final scatter of basil.
Serve right away with crusty bread to scoop up the creamy burrata and dressing.