I start by patting the chicken breasts dry with a paper towel, since dry chicken browns much better than wet chicken. I halve each breast horizontally into two thinner cutlets, then season both sides generously with salt and pepper. This step sets the foundation for a nicely seared, well-seasoned bite in every piece.
Next, I spread the flour on a plate and season it with a pinch of salt and pepper. I press each chicken piece into the flour, coating it lightly on both sides and shaking off any excess. This thin coating is what gives the chicken a golden crust and helps the sauce cling to it later.
I heat the butter and olive oil together in a large skillet over medium-high heat until the butter stops foaming. I lay the chicken in the pan without crowding it and let it cook undisturbed for about four minutes per side. I'm looking for a deep golden color before I flip, then I move the chicken to a plate once it's cooked through.
In the same skillet, I add the shallot and garlic, stirring for about a minute until they turn fragrant. I toss in the tomatoes and let them soften for a couple of minutes, releasing their juices into the pan. I pour in the chicken broth next, scraping up any browned bits from the bottom, since that's where a lot of the flavor lives.
I stir in the capers and red wine vinegar, then let the sauce simmer for two to three minutes until it thickens slightly. I return the chicken to the pan, spooning the sauce over each piece so it soaks in the flavor. A quick simmer here lets everything come together as one cohesive dish.
Right before serving, I scatter the torn basil leaves generously over the top. The residual heat from the pan releases the basil's aroma without wilting it into nothing. I like to finish with one more light crack of black pepper before bringing it straight to the table.