On a cutting board, slice each chicken breast in half horizontally to make 4-6 thin cutlets.
Season each side with salt and pepper.
Sprinkle the flour in an even layer on a plate and press each side of each chicken cutlet into the flour, lightly dusting off the excess.
Melt the butter in a large pan over medium high heat.
Once melted, add the chicken cutlets in batches, taking care to not overcrowd, and cook until golden on each side. Do not cook all the way through.
Remove the chicken to a plate and repeat with remaining pieces, adding more butter if needed.
Once the last piece of chicken is removed, drizzle the olive oil into the pan.
Once hot, add the cherry tomatoes and cook, undisturbed, until starting to turn golden brown and burst.
Stir, cooking occasionally, until the cherry tomatoes start to break down.
Stir in the garlic and continue to cook until fragrant, 30 seconds to 1 minute.
Stir in the oregano, paprika, salt, and pepper. Adjust seasonings to taste.
Pour in the wine and scrap the bottom of the pan, releasing any browned bits.
Nestle the seared chicken pieces back into the pan with the tomato sauce and simmer until the chicken is cooked through at the thickest part to 165˚F (74˚C).
Remove from heat, adjust seasonings to taste, and sprinkle with the chopped parsley or basil.
Serve the chicken with the tomato sauce immediately over pasta or rice.