Sprinkle the chicken breasts with salt and pepper. Wrap each piece with 2 large basil leaves, and then prosciutto, like an envelope. This should adhere fairly easily.
Place the eggs in a shallow bowl, and beat with 1 Tbsp water. In another shallow bowl, combine the panko with the grated parmesan. One at a time, dredge the chicken breasts through the beaten egg wash, and then the breadcrumb mixture, setting on a plate until ready to cook.
Heat a large cast iron skillet over medium heat, and add 2 Tbsp of olive oil with 1 Tbsp of butter - cook the chicken two cutlets at a time, flipping when golden brown, about 5 minutes per side. Finish the chicken and set aside to make the tomatoes. Add more olive oil and 1 Tbsp of butter for cooking the second round.
In the same skillet, add 2 Tbsp more of olive oil, and toss in the shallots, garlic and red pepper flakes, followed by the cherry tomatoes. Cook for several minutes until the tomatoes burst their skins and start to loosen a bit. Add the balsamic, a touch of honey, and the fresh thyme leaves. You should be left with a silky sauce surrounding the tomatoes.
To serve, plate each cutlet with a small mound of the tomatoes and sauce, and finish with fresh burrata and torn basil leaves. Enjoy!