Place a chicken breast between two pieces of plastic wrap or in a large zipper bag and flatten with a meat mallet until about 1/2 inch thick. Repeat with remaining breasts.

Season both sides of the chicken breasts with kosher salt and pepper.

Put the 1/4 cup flour in a shallow dish. Dredge each seasoned chicken breast in the flour, shaking off any excess.

Heat a large skillet over medium-high heat. Add 2 tablespoons of the 1/4 cup olive oil. When the oil is hot and shimmering, add the chicken breasts in a single layer (work in batches if needed). Sauté until lightly browned and cooked through, about 3–5 minutes per side depending on thickness. Transfer the cooked chicken to a plate and tent with foil to keep warm.

Add the remaining 2 tablespoons olive oil to the same skillet. Add the sliced garlic and the white parts of the scallions; sauté until fragrant and the scallion whites soften, about 1 minute.

Pour in the 1 cup chicken broth and use a wooden spoon to scrape up any browned bits from the bottom of the pan. Bring the broth to a gentle simmer.

Add the drained, halved artichoke hearts and the drained, sliced sun-dried tomatoes to the skillet. Simmer until the artichokes and tomatoes are heated through and the sauce has slightly reduced, about 2–3 minutes. Taste and add more salt and pepper if desired.

Return the cooked chicken breasts to the skillet, spooning the sauce over them. Simmer for 1–2 minutes more to warm the chicken through and meld the flavors.

Serve the chicken with the artichoke–tomato sauce spooned over the top and sprinkle with the remaining (green) parts of the sliced scallions.