Preheat oven to 350 degrees.
Heat the vegetable oil in your dutch oven or pan over medium high heat. Season the chicken breasts with salt & pepper; dredge in the flour, shaking off any excess. Working in batches if necessary, add the chicken to the pot/pan and cook until golden on both sides...about 5 minutes per side. Drain the chicken breasts on a paper towel lined plate.
Pour the oil out of the pot & carefully wipe it out with a paper towel. Add the 3 tablespoons of olive oil & heat over medium heat. Add the onion; season with salt & pepper to taste. Cook, stirring, until softened but not browned - about 4 minutes. Add the chicken broth, bring to a simmer, and cook until reduced by half, about 4 minutes. Add the tomato sauce and cook, stirring, for 3-4 minutes. Stir in the garlic, Italian seasoning, & red pepper flakes.
Return the chicken to the pot, and spoon the sauce over it. Add the peperoncini on top of the chicken & drizzle the peperoncini juice over the dish. Season with salt & pepper. Cover the pot (or if you don't have a dutch oven, use a tight fitting lid or 2 layers of aluminum foil) and cook for 35-40 minutes.