Pat chicken breast with paper towels to remove any excess moisture from packaging. Season chicken breast generously with salt and pepper.
Heat olive oil in a 12-inch skillet over medium-high heat, when it begins to shimmer, add chicken breast.
Cook, turning once, until firm to the touch, about 3-5 minutes per side, depending on the size of the breast. Transfer the chicken breasts to a plate and set aside.
Add the shallots and garlic to the same skillet. Cook over medium-low heat until the shallots are soft, about 4-6 minutes.
Add the balsamic vinegar and stir scrape up any brown bits on the bottom of the skillet, and cook until the vinegar is reduced and becomes syrupy.
Stir in the tomatoes, rosemary and chicken broth and bring to a boil. Reduce heat and simmer 8-10 minutes, until the tomatoes are soft and the sauce has thickened a bit.
Add butter and stir in until melted and combined.
Add chicken, along with any accumulated juices, and nestle chicken into the sauce. Simmer until chicken is warmed through. Flip chicken to coat with sauce.
Remove chicken to serving platter and sprinkle with chopped basil. Serve.