Whisk the olive oil, lemon juice, red-wine vinegar, minced garlic, parsley, dried basil, dried oregano, kosher salt, and black pepper together in a bowl. Pound thick chicken breasts to an even thickness before marinating so they cook at the same rate.

Place the chicken in a resealable bag or shallow dish and pour the marinade over it. Cover and refrigerate for at least 30 minutes, preferably 2–4 hours, or up to 8 hours; never marinate raw chicken at room temperature.

Bring a large pot of salted water to a boil. Cook the farfalle according to the package directions, usually 11–13 minutes, until al dente; reserve about 1/2 cup pasta water before draining.

Remove the chicken from the marinade and discard the used marinade. Pat the chicken lightly dry, then heat 1 tablespoon olive oil in a large skillet over medium-high heat.

Cook the chicken for approximately 5–7 minutes per side, depending on thickness, until browned and the thickest part reaches 165°F (74°C). Transfer it to a plate and let it rest for 5 minutes.

Add the remaining tablespoon of olive oil to the skillet. Cook the halved tomatoes for 2–3 minutes, add the chicken broth, scrape up the browned bits, and simmer for another 2–3 minutes.

Add the cooked farfalle and Parmesan to the skillet and toss until the pasta is coated. Add a little reserved pasta water if the sauce needs loosening, then taste and adjust with salt and pepper.

Slice the rested chicken and arrange it over the farfalle. Finish with chopped fresh basil or parsley and additional Parmesan before serving.