Pat the chicken breasts dry and, using a sharp knife held parallel to the cutting board, slice each breast horizontally to make 4 thin cutlets; season both sides with salt and pepper.
Heat 2 tablespoons olive oil in a large skillet over medium heat until shimmering.
Add the chicken cutlets and cook about 2 minutes per side until lightly browned and cooked through; transfer to a plate and cover with foil to keep warm.
Return the same pan to medium heat and add 2 tablespoons extra virgin olive oil.
Add the finely chopped onion and sauté 3–4 minutes until softened, then add the sliced garlic and cook 1 minute more without letting it brown.
Stir in 2 tablespoons tomato puree, 1 teaspoon sugar, and 1/2 tablespoon dried oregano; cook 1 minute to combine.
Add the halved cherry tomatoes and 1/2 cup chicken broth, bring to a simmer, and cook about 5 minutes until the tomatoes begin to break down.
Stir in the 1/4 cup basil leaves, then return the cooked chicken to the pan, spooning sauce over each piece (include any juices from the resting plate).
Let the chicken simmer in the sauce 2–3 minutes to warm through, then serve with extra basil if desired.