Preheat oven to 375 degrees. Preheat a large cast iron skillet over medium high heat.
Pat the chicken cutlets dry, and season liberally with kosher salt and black pepper - liberally! Place the flour in a shallow plate and dredge the cutlets through the flour to lightly coat, shaking off excess - you just want a very thin layer to help crust the chicken when you sear it.
Melt 2 Tbsp of butter and 2 Tbsp of Olive Oil in the skillet, and add the chicken breasts, cooking about 4-5 minutes each side until golden brown and just barely cooked through. You want to leave a little cooking time as you will finish the dish in the oven. This process will usually be done in two batches, as you want room around the chicken breasts so they sear instead of steaming.
After the chicken is cooked, put it all aside on a plate to make the sauce. Add the remaining 2 Tbsp of butter, the white wine, and thyme to the skillet, swirling to combine. Add the chicken back to the pan, spooning the sauce over each cutlet. Remove the pan from the heat, and add the tomatoes, scattering amidst the chicken and layering into the pan. Spoon the pesto around the chicken and tomatoes, and tear the burrata as well, arranging evenly. Top everything with some red pepper flakes, and throw it into the oven for a few minutes to roast the tomatoes and burrata.
Remove the pan from the oven, sprinkle heavily with freshly torn basil, and finish with flaky salt and a drizzle of runny honey if desired. Serve this hot chicken mess straight from the pan with a big spoon, and crusty bread on the side. The sauce is amazing to mop up.