In a bowl, toss together the chicken and pesto, coating the chicken fully in pesto. Combine the Panko and parmesan in a separate shallow bowl.Working with 1 cutlet at a time, press 2 basil leaves on top of each cutlet. Wrap 2 pieces of prosciutto around each cutlet to secure the basil. The oils from the pesto should moisten the prosciutto, if not drizzle lightly with olive oil. Dredge through the Panko, pressing to adhere. Place the chicken on a plate.To make the tomatoes. In a large skillet, over high heat, combine the olive oil, 2 cups tomatoes, shallots, garlic, thyme, lemon zest, and a pinch each of chili flakes, salt, and pepper. Cook until the tomatoes burst about, 10 minutes. Turn the heat off. Halve the remaining tomatoes and stir them in with the vinegar and honey. Heat a few tablespoons of oil in a large skillet over medium-high. When the oil shimmers, add the chicken and cook until golden brown, 3-4 minutes. Cook until golden brown, 3-4 minutes per side. Remove from the skillet and season with salt. Serve the chicken warm topped with torn burrata, tomatoes, and fresh basil.