Sprinkle chicken with salt. Place flour in a shallow bowl. Dredge each half chicken breast on both sides with flour, shaking off excess over bowl.
Warm 2 Tbsp. oil in a large skillet over medium-high heat. Add chicken and cook until golden on both sides, about 10 minutes total. Transfer to a plate and cover loosely with foil to keep warm.
Heat remaining 1 Tbsp. oil in skillet. Add onion and cook, stirring, until softened, about 5 minutes. Add broth and tomatoes and cook until sauce has thickened, about 3 minutes. Stir in olives and arugula and cook, stirring, until arugula has wilted, about 1 minute. Season with salt.
Pour sauce over chicken and serve.